{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "cell-00",
   "metadata": {},
   "source": "# Reinforcement learning with verifiable rewards\n\nSupervised fine-tuning is easy to explain: show the model the right answer and make that answer more likely. But for a lot of the problems we care about, nobody has written the right answer down. What we do have is a way to check one. Code either passes its tests or it doesn't, and a maths answer either matches or it doesn't.\n\nA check is enough to learn from. Let the model try a few times, check each try, and nudge it towards the ones that passed. That's reinforcement learning with verifiable rewards, RLVR for short, and it's the recipe behind DeepSeek-R1's reasoning ([DeepSeek-AI, 2025](https://arxiv.org/abs/2501.12948)).\n\n![The RLVR loop. The policy writes four replies to a GSM8K question, a checker compares each boxed answer with the reference, 72, and the update makes the replies that scored above the group's mean more likely.](https://raw.githubusercontent.com/AshishKumar4/dew/main/docs/assets/rlvr-loop.svg)\n\nThat picture is the whole method. We'll build it with GRPO, the variant DeepSeekMath introduced ([Shao et al., 2024](https://arxiv.org/abs/2402.03300)) and R1 was trained with, and use it to teach Qwen2.5-0.5B-Instruct grade-school maths with Dew's `GRPOObjective`."
  },
  {
   "cell_type": "markdown",
   "id": "cell-01",
   "metadata": {},
   "source": "## Setup"
  },
  {
   "cell_type": "code",
   "id": "cell-02",
   "metadata": {},
   "source": "%pip install -q \"dewml[cuda13,interop,streaming] @ git+https://github.com/AshishKumar4/dew@dea386da476a5b32f59b6bfef94993ace34f38c7\"",
   "execution_count": 1,
   "outputs": [
    {
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     "output_type": "stream",
     "text": "  Installing build dependencies ... \u001b[?25l\u001b[?25hdone\n"
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     "text": "  Getting requirements to build wheel ... \u001b[?25l\u001b[?25hdone\n"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "  Preparing metadata (pyproject.toml) ... \u001b[?25l\u001b[?25hdone\n"
    },
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    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "  Building wheel for dewml (pyproject.toml) ... \u001b[?25l\u001b[?25hdone\n"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "\u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\ncudf-cu13 26.6.0 requires pyarrow<24,>=19.0.0, but you have pyarrow 26.0.0 which is incompatible.\u001b[0m\u001b[31m\n\u001b[0m"
    }
   ]
  },
  {
   "cell_type": "code",
   "id": "cell-03",
   "metadata": {},
   "source": "MODEL = \"Qwen/Qwen2.5-0.5B-Instruct\"\nPROMPTS = 16          # questions per batch\nGROUPS = 8            # replies per question, G\nPROMPT_TOKENS = 256   # the longest question we keep, in tokens\nNEW_TOKENS = 512      # the longest reply\nSTEPS = 300           # batches of replies to train on\nLEARNING_RATE = 3e-6\nBETA = 0.04           # weight of the KL penalty\nSEED = 0",
   "execution_count": 2,
   "outputs": []
  },
  {
   "cell_type": "code",
   "id": "cell-04",
   "metadata": {},
   "source": "import json\nimport time\nfrom collections import Counter\nfrom dataclasses import replace\nfrom types import SimpleNamespace\n\nimport dew  # before JAX's backend opens, so its numerical settings apply\nimport jax\nimport jax.numpy as jnp\nimport numpy as np\nfrom datasets import load_dataset\n\nfrom dew import LocalTracker, Trainer, figures\nfrom dew.config import OptimConfig\nfrom dew.config.data import PromptData, loader\nfrom dew.data import HFTokenizer, Loading\nfrom dew.data.dataset import DataPartition\nfrom dew.data.prompts import INFO_KEY\nfrom dew.interop import PretrainedDecoder\nfrom dew.nn.backbones.decoder_block import remat_policy\nfrom dew.objectives.base import Step\nfrom dew.objectives.rl import GRPOObjective, PromptRecord, SampledRollout\nfrom dew.rl import clipped_surrogate_terms, k3_kl\nfrom dew.rl.sandbox import MathReward\nfrom dew.sampling import Sampling\nfrom dew.training import prepare_process\n\nnp.set_printoptions(precision=3, suppress=True)\nprepare_process()  # Dew's setup for a training process, before the first JAX call",
   "execution_count": 3,
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "id": "cell-05",
   "metadata": {},
   "source": "## A task we can check\n\nGSM8K ([Cobbe et al., 2021](https://arxiv.org/abs/2110.14168)) is a set of grade-school word problems, each with a worked solution that ends in a single number. We only need that number. We'll ask the model to show its working and put the final answer in `\\boxed{}`, and Dew's `MathReward` gives a reply 1 if its last boxed answer equals the reference and 0 otherwise. It compares numbers exactly, so `72`, `72.0` and `\\frac{144}{2}` all count.\n\nWhat happens when a reply gets the number right but forgets the box?"
  },
  {
   "cell_type": "code",
   "id": "cell-06",
   "metadata": {},
   "source": "gsm8k = load_dataset(\"openai/gsm8k\", \"main\")\ntrain, test = gsm8k[\"train\"], gsm8k[\"test\"]\n\n\ndef reference(row):\n    # A worked solution ends with \"#### 72\"; the number is all we keep.\n    return row[\"answer\"].split(\"####\")[-1].strip().replace(\",\", \"\")\n\n\nchecker = MathReward()\nquestion = train[0]\nprint(f\"{len(train)} training and {len(test)} test questions\\n\")\nprint(question[\"question\"], \"\\nreference:\", reference(question), \"\\n\")\nfor reply in [r\"In May she sold 48 / 2 = 24, so \\boxed{72} in all.\",\n              r\"She sold 48 * 2 = 96 in May, so \\boxed{144}.\",\n              r\"She sold 48 + 24 = 72 clips.\"]:\n    print(f\"{checker(reply, PromptRecord('gsm8k', reference(question))):.0f}  {reply}\")",
   "execution_count": 4,
   "outputs": [
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      "text/plain": "README.md:   0%|          | 0.00/7.93k [00:00<?, ?B/s]"
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    {
     "name": "stderr",
     "output_type": "stream",
     "text": "Warning: You are sending unauthenticated requests to the HF Hub. Please set a HF_TOKEN to enable higher rate limits and faster downloads.\n"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": "WARNING:huggingface_hub.utils._http:Warning: You are sending unauthenticated requests to the HF Hub. Please set a HF_TOKEN to enable higher rate limits and faster downloads.\n"
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      "text/plain": "main/train-00000-of-00001.parquet: reconstructing file:   0%|          |  0.00B / 2.31MB            "
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     "text": "7473 training and 1319 test questions\n\nNatalia sold clips to 48 of her friends in April, and then she sold half as many clips in May. How many clips did Natalia sell altogether in April and May? \nreference: 72 \n\n1  In May she sold 48 / 2 = 24, so \\boxed{72} in all.\n0  She sold 48 * 2 = 96 in May, so \\boxed{144}.\n0  She sold 48 + 24 = 72 clips.\n"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cell-07",
   "metadata": {},
   "source": "The unboxed 72 gets the same 0 as the wrong 144, because the checker only reads the answer inside the box. That's harsh, but it's what makes the check trustworthy: it never has to guess which number in a reply is the answer."
  },
  {
   "cell_type": "markdown",
   "id": "cell-08",
   "metadata": {},
   "source": "## The model before training\n\nWe start from Qwen2.5-0.5B-Instruct ([Qwen Team, 2024](https://arxiv.org/abs/2412.15115)), a small model that already follows instructions and can do some arithmetic. Before we train it, let's grade it on all 1,319 test questions, which it will never see during training. It answers greedily, always taking its most likely token, so the same question always gets the same reply."
  },
  {
   "cell_type": "code",
   "id": "cell-09",
   "metadata": {},
   "source": "source = PretrainedDecoder.load(MODEL, dtype=jnp.bfloat16, param_dtype=jnp.float32,\n                                max_seq_len=PROMPT_TOKENS + NEW_TOKENS)\ntokenizer = HFTokenizer(MODEL)\nstock = source.text_generation().sampling\nends = dict(eos_token_ids=stock.eos_token_ids, pad_token_id=stock.pad_token_id)\nINSTRUCTION = r\"Solve the problem step by step, then give the final answer in \\boxed{}.\"\n\n\ndef messages(row):\n    return [{\"role\": \"user\", \"content\": f\"{row['question']}\\n{INSTRUCTION}\"}]\n\n\ndef chat(row):\n    # The question as the model's chat template writes it, up to where the reply starts.\n    return tokenizer.tokenizer.apply_chat_template(messages(row), tokenize=False,\n                                                   add_generation_prompt=True)",
   "execution_count": 5,
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  {
   "cell_type": "code",
   "id": "cell-10",
   "metadata": {},
   "source": "def grade(variables, rows, batch=128):\n    # Every reply to `rows` from the model with these weights, and which ones are right.\n    task = replace(source, variables=variables).text_generation(\n        sampling=Sampling(temperature=0.0, **ends))\n    prompts = [chat(row) for row in rows]\n    replies = []\n    for start in range(0, len(prompts), batch):\n        replies += task(prompts[start:start + batch], NEW_TOKENS, key=SEED).text\n    right = [checker(reply, PromptRecord(\"gsm8k\", reference(row)))\n             for reply, row in zip(replies, rows)]\n    return replies, np.asarray(right)\n\n\nbefore, right_before = grade(source.variables, test)\nprint(f\"{right_before.mean():.1%} of the {len(test)} test questions right\\n\")\nprint(test[0][\"question\"], \"\\nreference:\", reference(test[0]), \"\\n\")\nprint(before[0])",
   "execution_count": 6,
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     "text": "48.7% of the 1319 test questions right\n\nJanet’s ducks lay 16 eggs per day. She eats three for breakfast every morning and bakes muffins for her friends every day with four. She sells the remainder at the farmers' market daily for $2 per fresh duck egg. How much in dollars does she make every day at the farmers' market? \nreference: 18 \n\nTo determine how much Janet makes at the farmers' market every day, we need to follow these steps:\n\n1. Calculate the total number of eggs laid by the ducks in a day.\n2. Determine how many eggs are sold at the farmers' market.\n3. Calculate the total revenue from selling the eggs.\n\n**Step 1: Calculate the total number of eggs laid by the ducks in a day.**\n\nJanet's ducks lay 16 eggs per day. Therefore, the total number of eggs laid in a day is:\n\\[ 16 \\text{ eggs/day} \\]\n\n**Step 2: Determine how many eggs are sold at the farmers' market.**\n\nJanet eats 3 eggs for breakfast every morning and bakes 4 muffins for her friends every day. Therefore, the total number of eggs she eats in a day is:\n\\[ 3 \\text{ eggs (breakfast)} + 4 \\text{ muffins (baking)} = 7 \\text{ eggs/day} \\]\n\nThe number of eggs she sells at the farmers' market is the total number of eggs she lays minus the number of eggs she eats:\n\\[ 16 \\text{ eggs} - 7 \\text{ eggs} = 9 \\text{ eggs/day} \\]\n\n**Step 3: Calculate the total revenue from selling the eggs.**\n\nJanet sells each egg for $2. Therefore, the total revenue from selling 9 eggs is:\n\\[ 9 \\text{ eggs} \\times 2 \\text{ dollars/egg} = 18 \\text{ dollars/day} \\]\n\nSo, the amount Janet makes at the farmers' market every day is:\n\\[ \\boxed{18} \\]\n"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cell-11",
   "metadata": {},
   "source": "Before any training it gets 48.7% of the test questions right. Its reply to the first one lays the problem out in steps, works through them, and boxes 18, which is right. On about half the questions it slips somewhere along the way, and on a few it never gets to a boxed answer at all. That's the sweet spot for RL: the model can already do the task some of the time, so when we sample several replies, some of them will pass."
  },
  {
   "cell_type": "markdown",
   "id": "cell-12",
   "metadata": {},
   "source": "## Why nudging towards the right replies works\n\nHow should a check, which only says yes or no, change the weights? Let's shrink the problem as far as it goes. Say the model had to pick one of five answers, and only one is right. Its policy is a softmax over five numbers, $\\pi_\\theta = \\mathrm{softmax}(\\theta)$, and we'd like to raise the expected reward $J(\\theta) = \\sum_a \\pi_\\theta(a)\\, R(a)$.\n\nBut a checker has no gradient; you can't differentiate \"the boxed number was 72\". There's a neat way around that. Because $\\nabla \\pi = \\pi\\, \\nabla \\log \\pi$, the gradient of the expected reward becomes an average over the model's own tries:\n\n$$\n\\nabla J = \\mathbb{E}_{a \\sim \\pi_\\theta}\\big[R(a)\\, \\nabla \\log \\pi_\\theta(a)\\big].\n$$\n\nRead it out loud: sample an answer, look at its reward, and push its log-probability up by that much. That's REINFORCE ([Williams, 1992](https://link.springer.com/article/10.1007/BF00992696)), and everything after it is a refinement of this one line. For the softmax the push is $\\nabla_\\theta \\log \\pi_\\theta(a) = \\mathrm{onehot}(a) - \\pi_\\theta$, which raises the sampled answer and lowers every answer in proportion to how likely it already was. A reply is the same kind of choice, made one token at a time.\n\nThe catch is that word \"average\". A reply costs hundreds of tokens to generate, so we only get a handful per question, and a handful of scores is a noisy estimate. If none of them is right, there's nothing to push up at all. No trick recovers a signal from tries that all failed, which is why RLVR needs questions the model sometimes gets right. But even when some tries do pass, comparing each try against its siblings tames a lot of the noise."
  },
  {
   "cell_type": "markdown",
   "id": "cell-13",
   "metadata": {},
   "source": "## Compare each reply with its group\n\n![Eight replies to one question scored 1 or 0, their mean, and the advantages. Right replies get a positive advantage and are pushed up, wrong ones get a negative one and are pushed down.](https://raw.githubusercontent.com/AshishKumar4/dew/main/docs/assets/rlvr-advantage.svg)\n\nSay a reply scores 1. How much should it go up? That depends on the question. On a question the model almost always gets right, one more right reply tells us little, and on one it usually fails, a right reply is exactly what we want more of. So GRPO samples a group of $G$ replies to the same question and judges each against the others, as in the picture: eight scores, their mean, and how far each one sits from the mean in units of the group's spread.\n\n$$\nA^{(i)} = \\frac{R^{(i)} - \\operatorname{mean}_j R^{(j)}}{\\operatorname{std}_j R^{(j)} + 10^{-6}}.\n$$\n\nBeat your group and you're pushed up, fall short and you're pushed down, and dividing by the spread puts easy and hard questions on one scale. When every reply in a group scores the same, all right or all wrong, every advantage is zero and the question teaches nothing this time. (The $10^{-6}$ only keeps us from dividing by zero.)\n\nHow does that look on real replies? Below, our untrained model gets eight tries at one training question. Each try is shown by the answer it boxed, or by its last few tokens if it never boxed one."
  },
  {
   "cell_type": "code",
   "id": "cell-14",
   "metadata": {},
   "source": "def record(index, row):\n    # A question as a prompt record: the chat, the reference the checker compares with,\n    # and the question's index, which reaches the checker as its extra information.\n    return {\"prompt\": messages(row), \"data_source\": \"gsm8k\",\n            \"reward_model\": {\"ground_truth\": reference(row)},\n            \"extra_info\": {\"index\": index}}\n\n\nwidth = PROMPT_TOKENS + NEW_TOKENS\n# Full recomputation in the backward pass keeps a step's activations small.\npolicy = source.model.clone(remat=remat_policy(\"full\"))\n# With beta > 0 the objective keeps a frozen copy of the starting weights for the KL.\nobjective = GRPOObjective(policy, width - 1, variables=source.variables, beta=BETA)\nsampling = Sampling(temperature=1.0, **ends)\nrollout = SampledRollout(objective, checker, decode=tokenizer.decode, groups=GROUPS,\n                         max_new_tokens=NEW_TOKENS, sampling=sampling)\nrecords = tuple(record(index, row) for index, row in enumerate(train))\ndata = loader(PromptData(source=records, tokenizer=MODEL, max_prompt_len=PROMPT_TOKENS,\n                         pad_id=stock.pad_token_id),\n              batch=PROMPTS, read=\"memory\", loading=Loading(workers=0))\nquestions = next(iter(data.train(DataPartition())))\nstart = SimpleNamespace(variables=source.variables, updates=0)\nbatch = rollout(start, questions, jax.random.key(SEED))",
   "execution_count": 7,
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  },
  {
   "cell_type": "code",
   "id": "cell-15",
   "metadata": {},
   "source": "# The batch packs replies into rows, several to a row where they fit. session_index\n# says which reply a token belongs to, and replies q * GROUPS to q * GROUPS + GROUPS - 1\n# answer the batch's question q.\ndef written(s):\n    return (batch[\"session_index\"] == s) & (batch[\"response_mask\"] > 0)\n\n\nsessions = range(PROMPTS * GROUPS)\nby_question = np.array([batch[\"advantages\"][written(s)][0] for s in sessions])\nby_question = by_question.reshape(PROMPTS, GROUPS)\n# The first question whose replies didn't all score the same.\nq = next(q for q in range(PROMPTS) if np.ptp(by_question[q]) > 0)\ngroup, advantages = range(q * GROUPS, (q + 1) * GROUPS), by_question[q]\n# A finished reply ends with the end-of-turn token, which the checker never sees.\nreplies = [[token for token in batch[\"input_ids\"][written(s)].tolist()\n            if token not in stock.eos_token_ids] for s in group]\ninfo = bytes(np.asarray(questions[INFO_KEY][q], np.uint8)).rstrip(b\"\\0\")\nasked = train[json.loads(info)[\"index\"]]\nrewards = np.array([checker(tokenizer.decode(reply), PromptRecord(\"gsm8k\", reference(asked)))\n                    for reply in replies])\nprint(asked[\"question\"], \"\\nreference:\", reference(asked))",
   "execution_count": 8,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "A math teacher had $100 to buy three different types of calculators. A basic calculator costs $8. A scientific calculator costs twice the price as the basic while a graphing calculator costs thrice the price as the scientific. How much change did she receive after buying those three different types of calculators? \nreference: 28\n"
    }
   ]
  },
  {
   "cell_type": "code",
   "id": "cell-16",
   "metadata": {},
   "source": "def answer(reply, shown=6):\n    # The tokens from the last \"boxed\" on, or the reply's last few without one.\n    pieces = [tokenizer.decode([token]) for token in reply]\n    boxed = [i for i, piece in enumerate(pieces) if \"boxed\" in piece]\n    start = boxed[-1] if boxed else len(pieces) - shown\n    return [piece.strip() or \"·\" for piece in pieces[start:start + shown]]\n\n\nshown = [answer(reply) for reply in replies]\nshade = [\"accent_fill\" if a > 0 else \"warm_fill\" for a in advantages]\nfigures.tokens(shown, fills=[[fill] * len(row) for fill, row in zip(shade, shown)],\n               notes=[f\"R {r:.0f}  A {a:+.2f}\" for r, a in zip(rewards, advantages)])",
   "execution_count": 9,
   "outputs": [
    {
     "data": {
      "text/html": "<svg xmlns=\"http://www.w3.org/2000/svg\" class=\"dew-figure\" viewBox=\"0 0 344.8 282\" width=\"344.8\" height=\"282\" role=\"img\" aria-label=\"token rows\"><title>token rows</title><style>.dew-figure{--fig-text:#16282b;--fig-muted:#4d6763;--fig-rule:#c9d9d5;--fig-surface:#f2f6f5;--fig-raised:#ffffff;--fig-accent:#0b7d71;--fig-accent-fill:#d3f1ec;--fig-warm:#b45f06;--fig-warm-fill:#f8e3c8;--fig-tint0:#cdeee8;--fig-tint1:#d6e6f5;--fig-tint2:#e6ddf3;--fig-tint3:#f5e6cf;--fig-sans:var(--sl-font,Geist, Inter, -apple-system, 'Segoe UI', Helvetica, Arial, sans-serif);--fig-mono:var(--sl-font-mono,'Geist Mono', 'JetBrains Mono', ui-monospace, Menlo, Consolas, monospace)}@media 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      "text/plain": "Figure(344.8 x 282, 8 shapes)"
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cell-17",
   "metadata": {},
   "source": "Five of the eight replies box 28, the reference, and score 1. Of the other three, one boxes 0, one boxes 18, and one runs out of room in the middle of a sum. With 5 of 8 right the group's mean is 0.625, so a right reply sits +0.72 above it in units of the spread and a wrong one −1.21 below. Notice the asymmetry: on a question the model mostly gets right, a miss gets pushed down harder than a hit gets pushed up. On a question it mostly fails, it would be the other way round."
  },
  {
   "cell_type": "markdown",
   "id": "cell-18",
   "metadata": {},
   "source": "## From one choice to a whole reply\n\n![One reply as the loss reads it: the end of the question with mask 0, and the tokens the policy wrote, 48 + 24 = 72, with mask 1, each carrying the reply's advantage.](https://raw.githubusercontent.com/AshishKumar4/dew/main/docs/assets/rlvr-sequence.svg)\n\nA reply is a sequence of choices, one per token, and its few hundred tokens have to share one reward. Because a reply's log-probability is the sum of its tokens' log-probabilities, we give every token the model wrote the same advantage, its reply's, as in the picture. The model didn't choose the question, and a mask $m$ (1 on written tokens, 0 everywhere else) leaves those tokens out. With $\\ell_t = \\log \\pi_\\theta(y_t \\mid q, y_{<t})$ for one token, Dew minimizes the average over every written token in the batch:\n\n$$\nL(\\theta) = -\\frac{\\sum_{i,t} m^{(i)}_t A^{(i)} \\ell^{(i)}_t}{\\sum_{i,t} m^{(i)}_t}.\n$$\n\nDoes Dew really compute this? We can check on the group above: the advantages straight from the checker's scores, and $L$ written out from Qwen's own next-token probabilities, its gradient against the gradient of Dew's loss."
  },
  {
   "cell_type": "code",
   "id": "cell-19",
   "metadata": {},
   "source": "# Without the division by the spread, centring on a mean that includes each reply's\n# own reward shrinks the expected gradient by (G - 1) / G without turning it;\n# dew.rl.rloo_advantage leaves the reply out.\nours = (rewards - rewards.mean()) / (rewards.std(ddof=1) + 1e-6)\nnp.testing.assert_allclose(advantages, ours, rtol=1e-5, atol=1e-6)\nprint(\"the advantages match\")",
   "execution_count": 10,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "the advantages match\n"
    }
   ]
  },
  {
   "cell_type": "code",
   "id": "cell-20",
   "metadata": {},
   "source": "# The rows that hold the group, with the loss kept to its replies.\nheld = np.isin(batch[\"session_index\"], group)\nrows = {name: column[held.any(axis=1)] for name, column in batch.items()}\nrows[\"response_mask\"] = rows[\"response_mask\"] * held[held.any(axis=1)]\nstep = Step(step=jnp.zeros((), jnp.int32), key=jax.random.key(0), ema=source.variables)\n\n\ndef relative_gap(a, b):\n    # How far apart two gradients are, as a share of the second one's length.\n    a, b = jax.tree.leaves(a), jax.tree.leaves(b)\n    gap = sum(float(jnp.sum((x - y) ** 2)) for x, y in zip(a, b))\n    return (gap / sum(float(jnp.sum(y ** 2)) for y in b)) ** 0.5",
   "execution_count": 11,
   "outputs": []
  },
  {
   "cell_type": "code",
   "id": "cell-21",
   "metadata": {},
   "source": "def dews_loss(params, model):\n    # Every written token rescored at the starting weights makes each ratio exactly 1,\n    # and the policy is still the reference, so neither the clip nor the KL moves it.\n    graded = GRPOObjective(model, width - 1, variables=source.variables, beta=BETA)\n    scored = rows | {\"old_log_probs\": graded.packed_log_probs(source.variables, rows)}\n    statistics, _ = graded.loss(params, scored, step)\n    return statistics.mean()[0]\n\n\ndef our_loss(params, model):\n    tokens = jnp.asarray(rows[\"input_ids\"])\n    m, a = jnp.asarray(rows[\"response_mask\"]), jnp.asarray(rows[\"advantages\"])\n    packing = {\"segment_ids\": jnp.asarray(rows[\"text_segment_ids\"])[:, :-1],\n               \"positions\": jnp.asarray(rows[\"text_positions\"])[:, :-1]}\n    logits = model.apply(params, tokens[:, :-1], **packing).astype(jnp.float32)\n    log_probs = jax.nn.log_softmax(logits)\n    ell = jnp.take_along_axis(log_probs, tokens[:, 1:, None], axis=-1)[..., 0]\n    return -jnp.sum(m[:, 1:] * a[:, 1:] * ell) / jnp.sum(m[:, 1:])\n\n\nfor dtype, bound in ((jnp.bfloat16, 0.02), (jnp.float32, 0.002)):\n    model = policy.clone(dtype=dtype)\n    gap = relative_gap(jax.grad(dews_loss)(source.variables, model),\n                       jax.grad(our_loss)(source.variables, model))\n    assert gap < bound\n    print(f\"{jnp.dtype(dtype).name}: the gradients differ by {gap:.2%} of their length\")",
   "execution_count": 12,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "bfloat16: the gradients differ by 1.04% of their length\n"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "float32: the gradients differ by 0.08% of their length\n"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cell-22",
   "metadata": {},
   "source": "Both match. In bfloat16, the precision the model computes in, the two gradients differ by 1.04% of their length; run in float32, the same comparison agrees to 0.08%. So the gap is rounding. bfloat16 keeps only two or three significant digits, and Dew's loss and ours do the same sums in a different order."
  },
  {
   "cell_type": "markdown",
   "id": "cell-23",
   "metadata": {},
   "source": "## Taking more than one step per batch\n\n![The clipped objective against the ratio. With a positive advantage it rises until the ratio reaches 1.2 and is flat after; with a negative one it is flat below 0.8 and past the dual clip at 3. Where it is flat, the clipped term gives the token no gradient.](https://raw.githubusercontent.com/AshishKumar4/dew/main/docs/assets/rlvr-clip.svg)\n\nGenerating replies is the slow part, so we'd like more than one gradient step per batch, or to train on replies a separate engine wrote while the last update ran. Either way the model has moved since it wrote the batch. Can we still train on it?\n\nPPO ([Schulman et al., 2017](https://arxiv.org/abs/1707.06347)) weights each token by the ratio $\\rho_t = \\pi_\\theta(y_t) / \\pi_\\text{old}(y_t)$, how much more or less likely the current model finds it than the model that wrote it, and stops pushing once that ratio has moved far enough the way the advantage asks:\n\n$$\n\\max\\big(-\\rho_t A,\\; -\\operatorname{clip}(\\rho_t, 0.8, 1.2)\\, A\\big).\n$$\n\nIn the picture, a positive advantage keeps improving the objective until the ratio reaches 1.2. Past that the line is flat, so while the ratio stays above 1.2 the token gets no gradient from the reward term. A negative advantage does the same below 0.8. For negative advantages Dew also caps the term at $-3A$, the dual clip of [Ye et al., 2020](https://arxiv.org/abs/1912.09729), so one token with a runaway ratio can't take over a step."
  },
  {
   "cell_type": "code",
   "id": "cell-24",
   "metadata": {},
   "source": "ratios = np.array([0.5, 0.9, 1.1, 1.5, 4.0])\nfor advantage in (1.0, -1.0):\n    formula = np.maximum(-ratios * advantage, -np.clip(ratios, 0.8, 1.2) * advantage)\n    if advantage < 0:\n        formula = np.minimum(formula, -3.0 * advantage)\n    terms, _ = clipped_surrogate_terms(jnp.log(ratios)[None], jnp.asarray([advantage]),\n                                       jnp.ones((1, ratios.size)))\n    np.testing.assert_allclose(terms[0], formula, rtol=1e-6)\n    print(f\"A = {advantage:+.0f}: terms {np.asarray(terms[0])} at ratios {ratios}\")",
   "execution_count": 13,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "A = +1: terms [-0.5 -0.9 -1.1 -1.2 -1.2] at ratios [0.5 0.9 1.1 1.5 4. ]\nA = -1: terms [0.8 0.9 1.1 1.5 3. ] at ratios [0.5 0.9 1.1 1.5 4. ]\n"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cell-25",
   "metadata": {},
   "source": "The terms follow the formula. With a positive advantage they stop at −1.2 past a ratio of 1.2, and with a negative one they sit at 0.8 below 0.8 and stop at 3 when the dual clip takes over."
  },
  {
   "cell_type": "markdown",
   "id": "cell-26",
   "metadata": {},
   "source": "## Keeping the model close to where it started\n\n![The k3 estimate of the KL divergence against the policy's probability over the reference's. It is zero where they agree and rises on both sides, so its gradient pulls a drifted token back.](https://raw.githubusercontent.com/AshishKumar4/dew/main/docs/assets/rlvr-kl.svg)\n\nGRPO's loss has one more term. While it chases the reward, the model can drift far from where it started and lose what it knew, so GRPO adds $\\beta$ times a per-token estimate of the KL divergence from the starting policy $\\pi_\\text{ref}$:\n\n$$\nk_3 = \\frac{\\pi_\\text{ref}}{\\pi_\\theta} - \\log \\frac{\\pi_\\text{ref}}{\\pi_\\theta} - 1 \\;\\ge 0.\n$$\n\nIt's zero when the two policies agree and grows as the model drifts either way, so its gradient points back towards the start, like the ball in the picture. Where the clip has flattened the reward term, this one can still pull, unless it has reached its own cap. Dew caps it at 10, as verl does, so one token the model has nearly ruled out can't swamp the loss."
  },
  {
   "cell_type": "code",
   "id": "cell-27",
   "metadata": {},
   "source": "policy_probs = np.array([0.1, 0.3, 0.5, 0.9, 0.01])\nreference_probs = np.array([0.3, 0.3, 0.25, 0.5, 0.2])\n# Dew caps k3 at 10, and the log-ratio inside it at ±20, as verl does.\nd = np.clip(np.log(reference_probs / policy_probs), -20, 20)\nestimate = k3_kl(jnp.log(policy_probs), jnp.log(reference_probs))\ncapped = np.minimum(np.exp(d) - d - 1, 10)\nnp.testing.assert_allclose(estimate, capped, rtol=1e-5, atol=1e-7)\nprint(\"formula:\", np.exp(d) - d - 1)\nprint(\"k3_kl:  \", np.asarray(estimate))",
   "execution_count": 14,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "formula: [ 0.901  0.     0.193  0.143 16.004]\nk3_kl:   [ 0.901  0.     0.193  0.143 10.   ]\n"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cell-28",
   "metadata": {},
   "source": "The first four values match the formula. For the last pair the formula gives 16.0, and Dew stops at the cap of 10."
  },
  {
   "cell_type": "markdown",
   "id": "cell-29",
   "metadata": {},
   "source": "## Training it\n\nNow we put the pieces together. Do right replies get more common? The mean reward is the share of sampled replies that pass the checker, so we can watch it, and also how many replies run out of room before they finish."
  },
  {
   "cell_type": "code",
   "id": "cell-30",
   "metadata": {},
   "source": "optimizer = OptimConfig(learning_rate=LEARNING_RATE, b2=0.99, weight_decay=0.0,\n                        clip_grads=1.0)\nrun = f\"runs/10-rlvr/{time.strftime('%Y%m%d-%H%M%S')}\"  # a fresh journal per run\ntrainer = Trainer(objective, optimizer, key=jax.random.key(SEED), rollout=rollout,\n                  tracker=LocalTracker(run))\nstate = trainer.fit(data, steps=STEPS, log_every=1)\n\nwith open(f\"{run}/scalars.jsonl\") as lines:\n    logged = [json.loads(line) for line in lines]\nlogged = [row for row in logged if \"rollout/reward/mean\" in row[\"scalars\"]]\nreward = [row[\"scalars\"][\"rollout/reward/mean\"] for row in logged]\ncut_off = [row[\"scalars\"][\"rollout/status/truncated\"] for row in logged]\nfigures.curve({\"mean reward\": reward, \"cut off\": cut_off},\n              x=[row[\"step\"] for row in logged], x_label=\"step\",\n              y_label=\"share of replies\", y_range=(0, 1), smooth=25)",
   "execution_count": 15,
   "outputs": [
    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">Training CausalTransformer from step 0 to 300: 494.0M parameters, on 1 × NVIDIA A100-SXM4-40GB, batch 16, float32 parameters, bfloat16 compute\n</pre>\n",
      "text/plain": "Training CausalTransformer from step 0 to 300: 494.0M parameters, on 1 × NVIDIA A100-SXM4-40GB, batch 16, float32 parameters, bfloat16 compute\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": "13:23:39 WARNING dew.nn.kernels.tuning: 6 kernel launches have no config measured on NVIDIA A100-SXM4-40GB, so they run the nearest measured shape's or their heuristic: measure them on a GPU to yourself with Trainer(tune_kernels=True) or `python tools/kernel_picks.py`\n"
    },
    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">train/mfu is unavailable for GRPOObjective: GRPOObjective needs counts for its auxiliary model forwards\n</pre>\n",
      "text/plain": "train/mfu is unavailable for GRPOObjective: GRPOObjective needs counts for its auxiliary model forwards\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">step 1/300  loss 0.04900  actor/pg_clipfrac 8.865e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 4.835e-04  kl 0  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.04900  data_wait_ms 0  grad_norm 0.8760  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 6,020  samples_per_sec 21.26  hlo_flops_util_lower_bound 23.6%  hfu_lower_bound 26.5%  unsized_kernels 291.0  rollout_seconds 27.61  accepted 100.0%  learning_rate 3.000e-06  length/mean 317.2  reward/mean 0.5391  status/truncated 6.2%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0\n</pre>\n",
      "text/plain": "step 1/300  loss 0.04900  actor/pg_clipfrac 8.865e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 4.835e-04  kl 0  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.04900  data_wait_ms 0  grad_norm 0.8760  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 6,020  samples_per_sec 21.26  hlo_flops_util_lower_bound 23.6%  hfu_lower_bound 26.5%  unsized_kernels 291.0  rollout_seconds 27.61  accepted 100.0%  learning_rate 3.000e-06  length/mean 317.2  reward/mean 0.5391  status/truncated 6.2%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">step 2/300  loss 0.02207  actor/pg_clipfrac 7.647e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 2.411e-04  kl 4.033e-04  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.02206  data_wait_ms 0.04483  grad_norm 0.7588  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 30,740  samples_per_sec 4.164  hlo_flops_util_lower_bound 4.6%  hfu_lower_bound 5.2%  unsized_kernels 291.0  rollout_seconds 11.65  accepted 100.0%  learning_rate 3.000e-06  length/mean 306.5  reward/mean 0.5312  status/truncated 3.9%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  1:09:45 left\n</pre>\n",
      "text/plain": "step 2/300  loss 0.02207  actor/pg_clipfrac 7.647e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 2.411e-04  kl 4.033e-04  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.02206  data_wait_ms 0.04483  grad_norm 0.7588  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 30,740  samples_per_sec 4.164  hlo_flops_util_lower_bound 4.6%  hfu_lower_bound 5.2%  unsized_kernels 291.0  rollout_seconds 11.65  accepted 100.0%  learning_rate 3.000e-06  length/mean 306.5  reward/mean 0.5312  status/truncated 3.9%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  1:09:45 left\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">step 3/300  loss 0.01741  actor/pg_clipfrac 8.513e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 2.674e-04  kl 6.922e-04  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.01739  data_wait_ms 0.02695  grad_norm 0.7917  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 16,883  samples_per_sec 7.581  hlo_flops_util_lower_bound 8.4%  hfu_lower_bound 9.4%  unsized_kernels 291.0  rollout_seconds 14.68  accepted 100.0%  learning_rate 3.000e-06  length/mean 330.4  reward/mean 0.2812  status/truncated 3.9%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  1:16:29 left\n</pre>\n",
      "text/plain": "step 3/300  loss 0.01741  actor/pg_clipfrac 8.513e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 2.674e-04  kl 6.922e-04  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.01739  data_wait_ms 0.02695  grad_norm 0.7917  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 16,883  samples_per_sec 7.581  hlo_flops_util_lower_bound 8.4%  hfu_lower_bound 9.4%  unsized_kernels 291.0  rollout_seconds 14.68  accepted 100.0%  learning_rate 3.000e-06  length/mean 330.4  reward/mean 0.2812  status/truncated 3.9%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  1:16:29 left\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">step 4/300  loss 0.005420  actor/pg_clipfrac 0.001291  actor/pg_clipfrac_lower 0  actor/ppo_kl 6.390e-04  kl 0.001437  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.005363  data_wait_ms 0.03191  grad_norm 0.9584  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 16,930  samples_per_sec 7.561  hlo_flops_util_lower_bound 8.4%  hfu_lower_bound 9.4%  unsized_kernels 291.0  rollout_seconds 14.70  accepted 100.0%  learning_rate 3.000e-06  length/mean 332.8  reward/mean 0.4609  status/truncated 7.8%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  1:18:35 left\n</pre>\n",
      "text/plain": "step 4/300  loss 0.005420  actor/pg_clipfrac 0.001291  actor/pg_clipfrac_lower 0  actor/ppo_kl 6.390e-04  kl 0.001437  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.005363  data_wait_ms 0.03191  grad_norm 0.9584  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 16,930  samples_per_sec 7.561  hlo_flops_util_lower_bound 8.4%  hfu_lower_bound 9.4%  unsized_kernels 291.0  rollout_seconds 14.70  accepted 100.0%  learning_rate 3.000e-06  length/mean 332.8  reward/mean 0.4609  status/truncated 7.8%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  1:18:35 left\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">step 5/300  loss 0.01116  actor/pg_clipfrac 9.689e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 5.173e-04  kl 0.001928  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.01108  data_wait_ms 0.02766  grad_norm 0.8565  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 16,796  samples_per_sec 7.621  hlo_flops_util_lower_bound 8.5%  hfu_lower_bound 9.5%  unsized_kernels 291.0  rollout_seconds 14.75  accepted 100.0%  learning_rate 3.000e-06  length/mean 330.6  reward/mean 0.4688  status/truncated 8.6%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  1:19:33 left\n</pre>\n",
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      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">step 299/300  loss 0.01311  actor/pg_clipfrac 9.982e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 4.219e-04  kl 0.05370  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.01096  data_wait_ms 0.02749  grad_norm 0.9304  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 16,792  samples_per_sec 7.623  hlo_flops_util_lower_bound 8.5%  hfu_lower_bound 9.5%  unsized_kernels 291.0  rollout_seconds 14.54  accepted 100.0%  learning_rate 3.000e-06  length/mean 273.9  reward/mean 0.5312  status/truncated 1.6%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  0:00:17 left\n</pre>\n",
      "text/plain": "step 299/300  loss 0.01311  actor/pg_clipfrac 9.982e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 4.219e-04  kl 0.05370  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg 0.01096  data_wait_ms 0.02749  grad_norm 0.9304  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 16,792  samples_per_sec 7.623  hlo_flops_util_lower_bound 8.5%  hfu_lower_bound 9.5%  unsized_kernels 291.0  rollout_seconds 14.54  accepted 100.0%  learning_rate 3.000e-06  length/mean 273.9  reward/mean 0.5312  status/truncated 1.6%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  0:00:17 left\n"
     },
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     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">step 300/300  loss -0.01090  actor/pg_clipfrac 9.158e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 3.467e-04  kl 0.04809  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg -0.01282  data_wait_ms 0.02608  grad_norm 0.9309  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 3,041  samples_per_sec 42.10  hlo_flops_util_lower_bound 46.8%  hfu_lower_bound 52.4%  unsized_kernels 291.0  rollout_seconds 14.62  accepted 100.0%  learning_rate 3.000e-06  length/mean 298.6  reward/mean 0.5781  status/truncated 3.9%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0\n</pre>\n",
      "text/plain": "step 300/300  loss -0.01090  actor/pg_clipfrac 9.158e-04  actor/pg_clipfrac_lower 0  actor/ppo_kl 3.467e-04  kl 0.04809  mismatch/ess 1.000  mismatch/k3_kl 0  mismatch/kl 0  pg -0.01282  data_wait_ms 0.02608  grad_norm 0.9309  compiles 1.000  cache_hits 0  cache_misses 0  peak_memory_gib 24.30  step_time_ms 3,041  samples_per_sec 42.10  hlo_flops_util_lower_bound 46.8%  hfu_lower_bound 52.4%  unsized_kernels 291.0  rollout_seconds 14.62  accepted 100.0%  learning_rate 3.000e-06  length/mean 298.6  reward/mean 0.5781  status/truncated 3.9%  accepted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,472  attempted/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 7,473  chunks_per_record/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0  rejected/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3/over_length 1.000  truncated/memory:6bc361644c6a032d8f5b721f528fdd7ea56e0305448c8749d34237c0effa90e3 0\n"
     },
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    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">Trained 300 steps in 1:25:49: first step after 117.86 s, then 0.1 step/s\n</pre>\n",
      "text/plain": "Trained 300 steps in 1:25:49: first step after 117.86 s, then 0.1 step/s\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">97.7% of the wall time in steps, final loss -0.01090\n</pre>\n",
      "text/plain": "97.7% of the wall time in steps, final loss -0.01090\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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style=\"fill:none;stroke:var(--fig-text);stroke-width:2.5;stroke-linejoin:round;stroke-linecap:round;stroke-dasharray:1;stroke-dashoffset:0;opacity:1;animation:fb0fa2bfd-3 8s ease-in-out infinite\"/><line x1=\"224\" y1=\"37\" x2=\"242\" y2=\"37\" style=\"stroke:var(--fig-text);stroke-width:2.5\"/><text x=\"248\" y=\"42\" font-size=\"15\" font-weight=\"400\" text-anchor=\"start\" style=\"fill:var(--fig-text);font-family:var(--fig-sans)\">cut off 0.04</text></svg>",
      "text/plain": "Figure(460 x 290, 23 shapes)"
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cell-31",
   "metadata": {},
   "source": "Each point is one batch of 16 questions, so the faint line jumps around a lot; the thick one is a running mean over 25 steps. It climbs from about 0.51 to a peak near 0.63 around step 150, then settles back to about 0.58. The share of replies cut off at 512 tokens rises to about 7% late in training, with single batches as high as 17%, and ends near 4%. But this reward is measured on the questions the model is learning from, sampled at temperature 1. Does the improvement carry over to questions it never trained on?"
  },
  {
   "cell_type": "markdown",
   "id": "cell-32",
   "metadata": {},
   "source": "## The model after training"
  },
  {
   "cell_type": "code",
   "id": "cell-33",
   "metadata": {},
   "source": "after, right_after = grade(state.variables, test)\nprint(f\"right before training {right_before.mean():.1%}, \"\n      f\"after {right_after.mean():.1%}\")\n\n\ndef outcomes(replies, right):\n    unboxed = np.mean([r\"\\boxed{\" not in reply for reply in replies])\n    return [right.mean(), 1 - right.mean() - unboxed, unboxed]\n\n\nfigures.bars([outcomes(before, right_before), outcomes(after, right_after)],\n             labels=[\"right\", \"wrong\", \"no answer\"],\n             captions=[\"before training\", \"after training\"],\n             colors=[\"accent\", \"warm\", \"tint1\"], y_label=\"share of test replies\")",
   "execution_count": 16,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "right before training 48.7%, after 52.1%\n"
    },
    {
     "data": {
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.dew-figure{--fig-text:#d8e5e2;--fig-muted:#93aba7;--fig-rule:#243639;--fig-surface:#0f191c;--fig-raised:#131f23;--fig-accent:#2fb8a8;--fig-accent-fill:#0b3531;--fig-warm:#f2b45a;--fig-warm-fill:#4a3416;--fig-tint0:#12433d;--fig-tint1:#1d3a52;--fig-tint2:#3f3358;--fig-tint3:#4f3a22}@media (prefers-reduced-motion:reduce){.dew-figure *{animation:none!important}}@keyframes f24cecc7c-0{0%{transform:scaleY(0.49)}3.75%{transform:scaleY(0.49)}70%{transform:scaleY(0.52)}92.5%{transform:scaleY(0.52)}100%{transform:scaleY(0.49)}}@keyframes f24cecc7c-1{0%{opacity:0}3.75%{opacity:0}5.62%{opacity:1}68.12%{opacity:1}70%{opacity:0}92.5%{opacity:0}100%{opacity:0}}@keyframes f24cecc7c-2{0%{opacity:0}70%{opacity:0}71.88%{opacity:1}92.5%{opacity:1}100%{opacity:0}}@keyframes f24cecc7c-3{0%{transform:scaleY(0.46)}3.75%{transform:scaleY(0.46)}70%{transform:scaleY(0.45)}92.5%{transform:scaleY(0.45)}100%{transform:scaleY(0.46)}}@keyframes f24cecc7c-4{0%{opacity:0}3.75%{opacity:0}5.62%{opacity:1}68.12%{opacity:1}70%{opacity:0}92.5%{opacity:0}100%{opacity:0}}@keyframes f24cecc7c-5{0%{opacity:0}70%{opacity:0}71.88%{opacity:1}92.5%{opacity:1}100%{opacity:0}}@keyframes f24cecc7c-6{0%{transform:scaleY(0.05)}3.75%{transform:scaleY(0.05)}70%{transform:scaleY(0.03)}92.5%{transform:scaleY(0.03)}100%{transform:scaleY(0.05)}}@keyframes f24cecc7c-7{0%{opacity:0}3.75%{opacity:0}5.62%{opacity:1}68.12%{opacity:1}70%{opacity:0}92.5%{opacity:0}100%{opacity:0}}@keyframes f24cecc7c-8{0%{opacity:0}70%{opacity:0}71.88%{opacity:1}92.5%{opacity:1}100%{opacity:0}}@keyframes f24cecc7c-9{0%{opacity:0}3.75%{opacity:0}5.62%{opacity:1}68.12%{opacity:1}70%{opacity:0}92.5%{opacity:0}100%{opacity:0}}@keyframes f24cecc7c-10{0%{opacity:0}70%{opacity:0}71.88%{opacity:1}92.5%{opacity:1}100%{opacity:0}}</style><line x1=\"16\" y1=\"220\" x2=\"444\" y2=\"220\" style=\"stroke:var(--fig-rule);stroke-width:1.25\"/><text x=\"16\" y=\"20\" font-size=\"15\" font-weight=\"400\" text-anchor=\"start\" style=\"fill:var(--fig-muted);font-family:var(--fig-sans)\">share of test replies</text><rect x=\"41.68\" y=\"48\" width=\"91.31\" height=\"172\" rx=\"3\" style=\"fill:var(--fig-accent);stroke:none;stroke-width:1;transform:scaleY(0.52);transform-box:fill-box;transform-origin:50% 100%;animation:f24cecc7c-0 8s ease-in-out infinite\"/><text x=\"87.33\" y=\"242\" font-size=\"15\" font-weight=\"400\" text-anchor=\"middle\" style=\"fill:var(--fig-text);font-family:var(--fig-mono)\">right</text><text x=\"87.33\" y=\"128.15\" font-size=\"14\" font-weight=\"400\" text-anchor=\"middle\" style=\"fill:var(--fig-muted);font-family:var(--fig-sans);opacity:0;animation:f24cecc7c-1 8s ease-in-out infinite\">0.49</text><text x=\"87.33\" y=\"122.41\" font-size=\"14\" font-weight=\"400\" text-anchor=\"middle\" style=\"fill:var(--fig-muted);font-family:var(--fig-sans);opacity:1;animation:f24cecc7c-2 8s ease-in-out infinite\">0.52</text><rect x=\"184.35\" y=\"48\" 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100%;animation:f24cecc7c-6 8s ease-in-out infinite\"/><text x=\"372.67\" y=\"242\" font-size=\"15\" font-weight=\"400\" text-anchor=\"middle\" style=\"fill:var(--fig-text);font-family:var(--fig-mono)\">no answer</text><text x=\"372.67\" y=\"203.39\" font-size=\"14\" font-weight=\"400\" text-anchor=\"middle\" style=\"fill:var(--fig-muted);font-family:var(--fig-sans);opacity:0;animation:f24cecc7c-7 8s ease-in-out infinite\">0.05</text><text x=\"372.67\" y=\"206.65\" font-size=\"14\" font-weight=\"400\" text-anchor=\"middle\" style=\"fill:var(--fig-muted);font-family:var(--fig-sans);opacity:1;animation:f24cecc7c-8 8s ease-in-out infinite\">0.03</text><text x=\"444\" y=\"20\" font-size=\"14\" font-weight=\"400\" text-anchor=\"end\" style=\"fill:var(--fig-muted);font-family:var(--fig-mono);opacity:0;animation:f24cecc7c-9 8s ease-in-out infinite\">before training</text><text x=\"444\" y=\"20\" font-size=\"14\" font-weight=\"400\" text-anchor=\"end\" style=\"fill:var(--fig-muted);font-family:var(--fig-mono);opacity:1;animation:f24cecc7c-10 8s ease-in-out infinite\">after training</text></svg>",
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     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ]
  },
  {
   "cell_type": "code",
   "id": "cell-34",
   "metadata": {},
   "source": "def outcome(reply, right):\n    return \"right\" if right else \"wrong\" if r\"\\boxed{\" in reply else \"no answer\"\n\n\nkinds = [\"right\", \"wrong\", \"no answer\"]\nmoves = Counter((outcome(*then), outcome(*now)) for then, now in\n                zip(zip(before, right_before), zip(after, right_after)))\nprint(f\"{'before, after':>14}\" + \"\".join(f\"{kind:>11}\" for kind in kinds))\nfor then in kinds:\n    print(f\"{then:>14}\" + \"\".join(f\"{moves[then, now]:>11}\" for now in kinds))",
   "execution_count": 17,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": " before, after      right      wrong  no answer\n         right        497        139          7\n         wrong        174        417         19\n     no answer         16         35         15\n"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cell-35",
   "metadata": {},
   "source": "Right answers go from 48.7% to 52.1%, but the table shows that this is a net change. Of the 643 questions the model got right before training, it now misses 146, and it picks up 190 it used to miss, 16 of them replies that never boxed an answer. Training moves questions both ways, and the bars show the difference."
  },
  {
   "cell_type": "code",
   "id": "cell-36",
   "metadata": {},
   "source": "fixed = next(i for i in range(len(test)) if right_after[i] > right_before[i])\nprint(test[fixed][\"question\"], \"\\nreference:\", reference(test[fixed]))\nprint(\"\\n--- before ---\\n\" + before[fixed])\nprint(\"\\n--- after ---\\n\" + after[fixed])",
   "execution_count": 18,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": "A robe takes 2 bolts of blue fiber and half that much white fiber.  How many bolts in total does it take? \nreference: 3\n\n--- before ---\nTo determine the total number of bolts of blue and white fiber needed, we start by defining the variables and setting up the equation based on the given information.\n\nLet:\n- \\( B \\) be the number of bolts of blue fiber.\n- \\( W \\) be the number of bolts of white fiber.\n\nAccording to the problem, the robe takes 2 bolts of blue fiber and half that much white fiber. Therefore, we can write the following equation:\n\n\\[ 2B = W \\]\n\nWe need to find the total number of bolts, which is the sum of the blue and white fibers:\n\n\\[ B + W \\]\n\nTo find \\( B + W \\), we can substitute \\( W \\) from the equation \\( 2B = W \\) into the total equation:\n\n\\[ B + W = B + 2B \\]\n\nSimplify the right-hand side:\n\n\\[ B + 2B = 3B \\]\n\nThus, the total number of bolts needed is:\n\n\\[ \\boxed{3B} \\]\n\n--- after ---\nTo determine the total number of bolts needed, we need to break down the problem step by step.\n\n1. **Identify the amount of blue fiber needed:**\n   - The problem states that it takes 2 bolts of blue fiber.\n\n2. **Determine the amount of white fiber needed:**\n   - The problem states that it takes half as much white fiber as blue fiber.\n   - Therefore, the amount of white fiber needed is \\( \\frac{2}{2} = 1 \\) bolt.\n\n3. **Calculate the total number of bolts needed:**\n   - To find the total number of bolts, we add the number of blue fiber bolts to the number of white fiber bolts.\n   - The total number of bolts is \\( 2 + 1 = 3 \\).\n\nThus, the total number of bolts needed is \\(\\boxed{3}\\).\n"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cell-37",
   "metadata": {},
   "source": "What went wrong before training? The model reversed the relation, writing $2B = W$ (twice as much white fiber, where the question says half), and boxed $3B$, which isn't even a number. After training it reads the numbers straight off the question: 2 bolts of blue, half of that is 1 bolt of white, 3 in all. All the model ever got back during training was a 1 or a 0 per reply."
  },
  {
   "cell_type": "markdown",
   "id": "cell-38",
   "metadata": {},
   "source": "## What changes at a larger scale\n\nThe loss stays the same. A bigger model gets its logits from a deeper network, but `GRPOObjective` computes the same per-token terms we just checked. What grows is everything around it. Rewards like running a program's tests need a sandbox (`dew.rl.sandbox`). While the trainer updates the model, a separate engine (Dew's `Server`, vLLM or SGLang) generates replies, so they can be one update old by the time we train on them. The example scores them again with the weights the update starts from, and caps each token's importance weight with `behavior_importance`. And long or multi-turn episodes are packed into rows with masks, which [tutorial 11](11-llms-in-an-agent-harness.ipynb) walks through.\n\n[`examples/train_rlvr.py`](https://github.com/AshishKumar4/dew/blob/main/examples/train_rlvr.py) puts all of this together and trains a model to write Python programs that pass their tests. [Post-training](../docs/concepts/post_training.md) covers Dew's other objectives."
  }
 ],
 "metadata": {
  "accelerator": "GPU",
  "colab": {
   "gpuType": "A100",
   "provenance": []
  },
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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   "name": "python"
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