LLM - 旋转位置编码 RoPE 代码详解

目录

一.引言

二.RoPE 理论

1.RoPE 矩阵形式

2.RoPE 图例形式

3.RoPE 实践分析

三.RoPE 代码分析

1.源码获取

2.源码分析

3.rotary_emb

3.1 __init__

3.2 forward

4.apply_rotary_pos_emb

4.1 rotate_half

4.2 apply_rotary_pos_emb

四.RoPE 代码实现

1.Q/K/V  States

2.rotary_emd

3.apply_rotray_pos_emd

五.总结

一.引言

LLM - 通俗理解位置编码与 RoPE 一文我们介绍了 RoPE 的理论基础和实现思路，下面我们再简单回顾下 RoPE 的实现思路并结合最新 LLM 模型的源代码看下 RoPE 具体是如何实现的。

二.RoPE 理论

1.RoPE 矩阵形式

结合内积的线性叠加性，我们可以将 2 维的旋转矩阵 R 推广到多维。相当于在原始 token 的向量上，两两截断通过 R 实现旋转操作，最后拼接为 R 维向量。犹如我们常用的向量维度都是偶数且是 2 的倍数，所以这里针对偶数向量：

由于上述 R 矩阵的稀疏性，使用矩阵乘法会造成算力的严重浪费，所以推荐使用下述线性方式计算 RoPE:

其中 ⊗ 是逐位对应相乘，即 Numpy、Tensorflow 等计算框架中的∗运算。从这个实现也可以看到，RoPE 可以视为是乘性位置编码的变体，将多个二维旋转的叠加：

2.RoPE 图例形式

上面将二维旋转矩阵 R 推广到多维旋转矩阵，并基于其矩阵特点，将稀疏矩阵计算简化至对位相乘，通过下图可以更清晰地理解计算过程：

因为上面的  是两两表示的，所以实际计算中，也是将 token 词向量两两分成，这也解释了为什么公式里到处是 d/2。其添加位置信息步骤如下：

- 遍历每个位置的 token，位置为 m，范围为 Query / Key 的长度

- 获取对应 token Embedding，dim = d

- [2i: 2i+1] for i in range(0, d/2) 两两获取向量，根据公式计算 θ 并通过 R 进行旋转变换

- 得到新的 Position Encoded Query / Key

-  Attention 操作并自动引入 (m-n) 的相对距离信息

从右上角的图中也可以更直观的看到向量旋转的过程。

3.RoPE 实践分析

观察上面的图我们可以看到 RoPE 计算主要分两部分

◆ 旋转角度 θ

这里 theta 完全由 Q、K、V 的向量长度 d 决定，即 dim。

◆ 位置 Position m

位置 m 对应我们的 query 长度，实际代码中由 max_position_embeddings 参数决定，可以理解为模型支持的最长 query 的长度，因此 max 有了，m 的范围也就有了。

◆ Freq 频次矩阵

结合上面的信息，针对一个固定了最长 query 长度 m 和向量维度 d 的 LLM，我们可以提前将其对应的旋转变换矩阵构造完成：

θ 有的地方写 [1, d/2]，有的地方范围写 [0, d/2-1]；其次位置 Position 有的地方写 [0, m-1]，有的写 [1, m]，这里写成从 1 开头为了方便大家理解，实际代码中会从 0 开始。结合这个 Rd 的变换矩阵，分别执行 cos 和 sin，便可以得到我们计算所需的全位置全维度的变换矩阵。下面看下代码如何实现。

三.RoPE 代码分析

1.源码获取

开始分析前，我们首先需要搞清楚源代码在哪，然后再开始分析。InternLM-20B 是最近最新推出的新版大模型，我们找最新模型的源码看下对应 RoPE Embedding 部分怎么执行。

◆ Hugging Face

在搜索框搜索对应模型名称。

◆ InternLM-20B

点击模型的 Files and versions 选项，对应的 modeling_xxx.py 即为实现源码。

◆ modeling_internlm.py

一共 999 行，粘过来省的大家再去 HF 上翻了。

# coding=utf-8
# Copyright 2022 EleutherAI and the HuggingFace Inc. team. All rights reserved.
#
# This code is based on EleutherAI's GPT-NeoX library and the GPT-NeoX
# and OPT implementations in this library. It has been modified from its
# original forms to accommodate minor architectural differences compared
# to GPT-NeoX and OPT used by the Meta AI team that trained the model.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
""" PyTorch InternLM model."""
import math
from typing import List, Optional, Tuple, Union
import threading, queue
 
import torch
import torch.utils.checkpoint
from torch import nn
from torch.nn import BCEWithLogitsLoss, CrossEntropyLoss, MSELoss
 
from transformers.activations import ACT2FN
from transformers.modeling_outputs import BaseModelOutputWithPast, CausalLMOutputWithPast, SequenceClassifierOutputWithPast
from transformers.modeling_utils import PreTrainedModel
from transformers.generation.streamers import BaseStreamer
from transformers.utils import add_start_docstrings, add_start_docstrings_to_model_forward, logging, replace_return_docstrings
from .configuration_internlm import InternLMConfig
 
 
logger = logging.get_logger(__name__)
 
_CONFIG_FOR_DOC = "InternLMConfig"
 
# TODO: https://bitddd.blog.csdn.net/article/details/133174206
# Copied from transformers.models.bart.modeling_bart._make_causal_mask
def _make_causal_mask(
    input_ids_shape: torch.Size, dtype: torch.dtype, device: torch.device, past_key_values_length: int = 0
):
    """
    Make causal mask used for bi-directional self-attention.
    """
    bsz, tgt_len = input_ids_shape
    mask = torch.full((tgt_len, tgt_len), torch.tensor(torch.finfo(dtype).min, device=device), device=device)
    mask_cond = torch.arange(mask.size(-1), device=device)
    mask.masked_fill_(mask_cond < (mask_cond + 1).view(mask.size(-1), 1), 0)
    mask = mask.to(dtype)
 
    if past_key_values_length > 0:
        mask = torch.cat([torch.zeros(tgt_len, past_key_values_length, dtype=dtype, device=device), mask], dim=-1)
    return mask[None, None, :, :].expand(bsz, 1, tgt_len, tgt_len + past_key_values_length)
 
 
# Copied from transformers.models.bart.modeling_bart._expand_mask
def _expand_mask(mask: torch.Tensor, dtype: torch.dtype, tgt_len: Optional[int] = None):
    """
    Expands attention_mask from `[bsz, seq_len]` to `[bsz, 1, tgt_seq_len, src_seq_len]`.
    """
    bsz, src_len = mask.size()
    tgt_len = tgt_len if tgt_len is not None else src_len
 
    expanded_mask = mask[:, None, None, :].expand(bsz, 1, tgt_len, src_len).to(dtype)
 
    inverted_mask = 1.0 - expanded_mask
 
    return inverted_mask.masked_fill(inverted_mask.to(torch.bool), torch.finfo(dtype).min)
 
 
class InternLMRMSNorm(nn.Module):
    def __init__(self, hidden_size, eps=1e-6):
        """
        InternLMRMSNorm is equivalent to T5LayerNorm
        """
        super().__init__()
        self.weight = nn.Parameter(torch.ones(hidden_size))
        self.variance_epsilon = eps
 
    def forward(self, hidden_states):
        variance = hidden_states.to(torch.float32).pow(2).mean(-1, keepdim=True)
        hidden_states = hidden_states * torch.rsqrt(variance + self.variance_epsilon)
 
        # convert into half-precision if necessary
        if self.weight.dtype in [torch.float16, torch.bfloat16]:
            hidden_states = hidden_states.to(self.weight.dtype)
 
        return self.weight * hidden_states
 
 
class InternLMRotaryEmbedding(torch.nn.Module):
    def __init__(self, dim, max_position_embeddings=2048, base=10000, device=None):
        super().__init__()
        inv_freq = 1.0 / (base ** (torch.arange(0, dim, 2).float().to(device) / dim))
        self.register_buffer("inv_freq", inv_freq, persistent=False)
 
        # Build here to make `torch.jit.trace` work.
        self.max_seq_len_cached = max_position_embeddings
        t = torch.arange(self.max_seq_len_cached, device=self.inv_freq.device, dtype=self.inv_freq.dtype)
        freqs = torch.einsum("i,j->ij", t, self.inv_freq)
        # Different from paper, but it uses a different permutation in order to obtain the same calculation
        emb = torch.cat((freqs, freqs), dim=-1)
        self.register_buffer("cos_cached", emb.cos()[None, None, :, :], persistent=False)
        self.register_buffer("sin_cached", emb.sin()[None, None, :, :], persistent=False)
 
    def forward(self, x, seq_len=None):
        # x: [bs, num_attention_heads, seq_len, head_size]
        # This `if` block is unlikely to be run after we build sin/cos in `__init__`. Keep the logic here just in case.
        if seq_len > self.max_seq_len_cached:
            self.max_seq_len_cached = seq_len
            t = torch.arange(self.max_seq_len_cached, device=x.device, dtype=self.inv_freq.dtype)
            freqs = torch.einsum("i,j->ij", t, self.inv_freq)
            # Different from paper, but it uses a different permutation in order to obtain the same calculation
            emb = torch.cat((freqs, freqs), dim=-1).to(x.device)
            self.register_buffer("cos_cached", emb.cos()[None, None, :, :], persistent=False)
            self.register_buffer("sin_cached", emb.sin()[None, None, :, :], persistent=False)
        return (
            self.cos_cached[:, :, :seq_len, ...].to(dtype=x.dtype),
            self.sin_cached[:, :, :seq_len, ...].to(dtype=x.dtype),
        )
 
 
def rotate_half(x):
    """Rotates half the hidden dims of the input."""
    x1 = x[..., : x.shape[-1] // 2]
    x2 = x[..., x.shape[-1] // 2 :]
    return torch.cat((-x2, x1), dim=-1)
 
 
def apply_rotary_pos_emb(q, k, cos, sin, position_ids):
    # The first two dimensions of cos and sin are always 1, so we can `squeeze` them.
    cos = cos.squeeze(1).squeeze(0)  # [seq_len, dim]
    sin = sin.squeeze(1).squeeze(0)  # [seq_len, dim]
    cos = cos[position_ids].unsqueeze(1)  # [bs, 1, seq_len, dim]
    sin = sin[position_ids].unsqueeze(1)  # [bs, 1, seq_len, dim]
    q_embed = (q * cos) + (rotate_half(q) * sin)
    k_embed = (k * cos) + (rotate_half(k) * sin)
    return q_embed, k_embed
 
 
class InternLMMLP(nn.Module):
    def __init__(
        self,
        hidden_size: int,
        intermediate_size: int,
        hidden_act: str,
    ):
        super().__init__()
        self.gate_proj = nn.Linear(hidden_size, intermediate_size, bias=False)
        self.down_proj = nn.Linear(intermediate_size, hidden_size, bias=False)
        self.up_proj = nn.Linear(hidden_size, intermediate_size, bias=False)
        self.act_fn = ACT2FN[hidden_act]
 
    def forward(self, x):
        return self.down_proj(self.act_fn(self.gate_proj(x)) * self.up_proj(x))
 
 
class InternLMAttention(nn.Module):
    """Multi-headed attention from 'Attention Is All You Need' paper"""
 
    def __init__(self, config: InternLMConfig):
        super().__init__()
        self.config = config
        self.hidden_size = config.hidden_size
        self.num_heads = config.num_attention_heads
        self.head_dim = self.hidden_size // self.num_heads
        self.max_position_embeddings = config.max_position_embeddings
 
        if (self.head_dim * self.num_heads) != self.hidden_size:
            raise ValueError(
                f"hidden_size must be divisible by num_heads (got `hidden_size`: {self.hidden_size}"
                f" and `num_heads`: {self.num_heads})."
            )
        self.q_proj = nn.Linear(self.hidden_size, self.num_heads * self.head_dim, bias=config.bias)
        self.k_proj = nn.Linear(self.hidden_size, self.num_heads * self.head_dim, bias=config.bias)
        self.v_proj = nn.Linear(self.hidden_size, self.num_heads * self.head_dim, bias=config.bias)
        self.o_proj = nn.Linear(self.num_heads * self.head_dim, self.hidden_size, bias=config.bias)
        self.rotary_emb = InternLMRotaryEmbedding(self.head_dim, max_position_embeddings=self.max_position_embeddings)
 
    def _shape(self, tensor: torch.Tensor, seq_len: int, bsz: int):
        return tensor.view(bsz, seq_len, self.num_heads, self.head_dim).transpose(1, 2).contiguous()
 
    def forward(
        self,
        hidden_states: torch.Tensor,
        attention_mask: Optional[torch.Tensor] = None,
        position_ids: Optional[torch.LongTensor] = None,
        past_key_value: Optional[Tuple[torch.Tensor]] = None,
        output_attentions: bool = False,
        use_cache: bool = False,
    ) -> Tuple[torch.Tensor, Optional[torch.Tensor], Optional[Tuple[torch.Tensor]]]:
        bsz, q_len, _ = hidden_states.size()
 
        query_states = self.q_proj(hidden_states).view(bsz, q_len, self.num_heads, self.head_dim).transpose(1, 2)
        key_states = self.k_proj(hidden_states).view(bsz, q_len, self.num_heads, self.head_dim).transpose(1, 2)
        value_states = self.v_proj(hidden_states).view(bsz, q_len, self.num_heads, self.head_dim).transpose(1, 2)
 
        kv_seq_len = key_states.shape[-2]
        if past_key_value is not None:
            kv_seq_len += past_key_value[0].shape[-2]
        cos, sin = self.rotary_emb(value_states, seq_len=kv_seq_len)
        query_states, key_states = apply_rotary_pos_emb(query_states, key_states, cos, sin, position_ids)
        # [bsz, nh, t, hd]
 
        if past_key_value is not None:
            # reuse k, v, self_attention
            key_states = torch.cat([past_key_value[0], key_states], dim=2)
            value_states = torch.cat([past_key_value[1], value_states], dim=2)
 
        past_key_value = (key_states, value_states) if use_cache else None
 
        attn_weights = torch.matmul(query_states, key_states.transpose(2, 3)) / math.sqrt(self.head_dim)
 
        if attn_weights.size() != (bsz, self.num_heads, q_len, kv_seq_len):
            raise ValueError(
                f"Attention weights should be of size {(bsz, self.num_heads, q_len, kv_seq_len)}, but is"
                f" {attn_weights.size()}"
            )
 
        if attention_mask is not None:
            if attention_mask.size() != (bsz, 1, q_len, kv_seq_len):
                raise ValueError(
                    f"Attention mask should be of size {(bsz, 1, q_len, kv_seq_len)}, but is {attention_mask.size()}"
                )
            attn_weights = attn_weights + attention_mask
            attn_weights = torch.max(attn_weights, torch.tensor(torch.finfo(attn_weights.dtype).min))
 
        # upcast attention to fp32
        attn_weights = nn.functional.softmax(attn_weights, dim=-1, dtype=torch.float32).to(query_states.dtype)
        attn_output = torch.matmul(attn_weights, value_states)
 
        if attn_output.size() != (bsz, self.num_heads, q_len, self.head_dim):
            raise ValueError(
                f"`attn_output` should be of size {(bsz, self.num_heads, q_len, self.head_dim)}, but is"
                f" {attn_output.size()}"
            )
 
        attn_output = attn_output.transpose(1, 2)
        attn_output = attn_output.reshape(bsz, q_len, self.hidden_size)
 
        attn_output = self.o_proj(attn_output)
 
        if not output_attentions:
            attn_weights = None
 
        return attn_output, attn_weights, past_key_value
 
 
class InternLMDecoderLayer(nn.Module):
    def __init__(self, config: InternLMConfig):
        super().__init__()
        self.hidden_size = config.hidden_size
        self.self_attn = InternLMAttention(config=config)
        self.mlp = InternLMMLP(
            hidden_size=self.hidden_size,
            intermediate_size=config.intermediate_size,
            hidden_act=config.hidden_act,
        )
        self.input_layernorm = InternLMRMSNorm(config.hidden_size, eps=config.rms_norm_eps)
        self.post_attention_layernorm = InternLMRMSNorm(config.hidden_size, eps=config.rms_norm_eps)
 
    def forward(
        self,
        hidden_states: torch.Tensor,
        attention_mask: Optional[torch.Tensor] = None,
        position_ids: Optional[torch.LongTensor] = None,
        past_key_value: Optional[Tuple[torch.Tensor]] = None,
        output_attentions: Optional[bool] = False,
        use_cache: Optional[bool] = False,
    ) -> Tuple[torch.FloatTensor, Optional[Tuple[torch.FloatTensor, torch.FloatTensor]]]:
        """
        Args:
            hidden_states (`torch.FloatTensor`): input to the layer of shape `(batch, seq_len, embed_dim)`
            attention_mask (`torch.FloatTensor`, *optional*): attention mask of size
                `(batch, 1, tgt_len, src_len)` where padding elements are indicated by very large negative values.
            output_attentions (`bool`, *optional*):
                Whether or not to return the attentions tensors of all attention layers. See `attentions` under
                returned tensors for more detail.
            use_cache (`bool`, *optional*):
                If set to `True`, `past_key_values` key value states are returned and can be used to speed up decoding
                (see `past_key_values`).
            past_key_value (`Tuple(torch.FloatTensor)`, *optional*): cached past key and value projection states
        """
 
        residual = hidden_states
 
        hidden_states = self.input_layernorm(hidden_states)
 
        # Self Attention
        hidden_states, self_attn_weights, present_key_value = self.self_attn(
            hidden_states=hidden_states,
            attention_mask=attention_mask,
            position_ids=position_ids,
            past_key_value=past_key_value,
            output_attentions=output_attentions,
            use_cache=use_cache,
        )
        hidden_states = residual + hidden_states
 
        # Fully Connected
        residual = hidden_states
        hidden_states = self.post_attention_layernorm(hidden_states)
        hidden_states = self.mlp(hidden_states)
        hidden_states = residual + hidden_states
 
        outputs = (hidden_states,)
 
        if output_attentions:
            outputs += (self_attn_weights,)
 
        if use_cache:
            outputs += (present_key_value,)
 
        return outputs
 
 
INTERNLM_START_DOCSTRING = r"""
    This model inherits from [`PreTrainedModel`]. Check the superclass documentation for the generic methods the
    library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads
    etc.)
    This model is also a PyTorch [torch.nn.Module](https://pytorch.org/docs/stable/nn.html#torch.nn.Module) subclass.
    Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage
    and behavior.
    Parameters:
        config ([`InternLMConfig`]):
            Model configuration class with all the parameters of the model. Initializing with a config file does not
            load the weights associated with the model, only the configuration. Check out the
            [`~PreTrainedModel.from_pretrained`] method to load the model weights.
"""
 
 
@add_start_docstrings(
    "The bare InternLM Model outputting raw hidden-states without any specific head on top.",
    INTERNLM_START_DOCSTRING,
)
class InternLMPreTrainedModel(PreTrainedModel):
    config_class = InternLMConfig
    base_model_prefix = "model"
    supports_gradient_checkpointing = True
    _no_split_modules = ["InternLMDecoderLayer"]
    _keys_to_ignore_on_load_unexpected = [r"decoder\.version"]
 
    def _init_weights(self, module):
        std = self.config.initializer_range
        if isinstance(module, nn.Linear):
            module.weight.data.normal_(mean=0.0, std=std)
            if module.bias is not None:
                module.bias.data.zero_()
        elif isinstance(module, nn.Embedding):
            module.weight.data.normal_(mean=0.0, std=std)
            if module.padding_idx is not None:
                module.weight.data[module.padding_idx].zero_()
 
    def _set_gradient_checkpointing(self, module, value=False):
        if isinstance(module, InternLMModel):
            module.gradient_checkpointing = value
 
 
INTERNLM_INPUTS_DOCSTRING = r"""
    Args:
        input_ids (`torch.LongTensor` of shape `(batch_size, sequence_length)`):
            Indices of input sequence tokens in the vocabulary. Padding will be ignored by default should you provide
            it.
            Indices can be obtained using [`AutoTokenizer`]. See [`PreTrainedTokenizer.encode`] and
            [`PreTrainedTokenizer.__call__`] for details.
            [What are input IDs?](../glossary#input-ids)
        attention_mask (`torch.Tensor` of shape `(batch_size, sequence_length)`, *optional*):
            Mask to avoid performing attention on padding token indices. Mask values selected in `[0, 1]`:
            - 1 for tokens that are **not masked**,
            - 0 for tokens that are **masked**.
            [What are attention masks?](../glossary#attention-mask)
            Indices can be obtained using [`AutoTokenizer`]. See [`PreTrainedTokenizer.encode`] and
            [`PreTrainedTokenizer.__call__`] for details.
            If `past_key_values` is used, optionally only the last `decoder_input_ids` have to be input (see
            `past_key_values`).
            If you want to change padding behavior, you should read [`modeling_opt._prepare_decoder_attention_mask`]
            and modify to your needs. See diagram 1 in [the paper](https://arxiv.org/abs/1910.13461) for more
            information on the default strategy.
            - 1 indicates the head is **not masked**,
            - 0 indicates the head is **masked**.
        position_ids (`torch.LongTensor` of shape `(batch_size, sequence_length)`, *optional*):
            Indices of positions of each input sequence tokens in the position embeddings. Selected in the range `[0,
            config.n_positions - 1]`.
            [What are position IDs?](../glossary#position-ids)
        past_key_values (`tuple(tuple(torch.FloatTensor))`, *optional*, returned when `use_cache=True` is passed or when `config.use_cache=True`):
            Tuple of `tuple(torch.FloatTensor)` of length `config.n_layers`, with each tuple having 2 tensors of shape
            `(batch_size, num_heads, sequence_length, embed_size_per_head)`) and 2 additional tensors of shape
            `(batch_size, num_heads, encoder_sequence_length, embed_size_per_head)`.
            Contains pre-computed hidden-states (key and values in the self-attention blocks and in the cross-attention
            blocks) that can be used (see `past_key_values` input) to speed up sequential decoding.
            If `past_key_values` are used, the user can optionally input only the last `decoder_input_ids` (those that
            don't have their past key value states given to this model) of shape `(batch_size, 1)` instead of all
            `decoder_input_ids` of shape `(batch_size, sequence_length)`.
        inputs_embeds (`torch.FloatTensor` of shape `(batch_size, sequence_length, hidden_size)`, *optional*):
            Optionally, instead of passing `input_ids` you can choose to directly pass an embedded representation. This
            is useful if you want more control over how to convert `input_ids` indices into associated vectors than the
            model's internal embedding lookup matrix.
        use_cache (`bool`, *optional*):
            If set to `True`, `past_key_values` key value states are returned and can be used to speed up decoding (see
            `past_key_values`).
        output_attentions (`bool`, *optional*):
            Whether or not to return the attentions tensors of all attention layers. See `attentions` under returned
            tensors for more detail.
        output_hidden_states (`bool`, *optional*):
            Whether or not to return the hidden states of all layers. See `hidden_states` under returned tensors for
            more detail.
        return_dict (`bool`, *optional*):
            Whether or not to return a [`~utils.ModelOutput`] instead of a plain tuple.
"""
 
 
@add_start_docstrings(
    "The bare InternLM Model outputting raw hidden-states without any specific head on top.",
    INTERNLM_START_DOCSTRING,
)
class InternLMModel(InternLMPreTrainedModel):
    """
    Transformer decoder consisting of *config.num_hidden_layers* layers. Each layer is a [`InternLMDecoderLayer`]
    Args:
        config: InternLMConfig
    """
    _auto_class = "AutoModel"
 
    def __init__(self, config: InternLMConfig):
        super().__init__(config)
        self.padding_idx = config.pad_token_id
        self.vocab_size = config.vocab_size
 
        self.embed_tokens = nn.Embedding(config.vocab_size, config.hidden_size, self.padding_idx)
        self.layers = nn.ModuleList([InternLMDecoderLayer(config) for _ in range(config.num_hidden_layers)])
        self.norm = InternLMRMSNorm(config.hidden_size, eps=config.rms_norm_eps)
 
        self.gradient_checkpointing = False
        # Initialize weights and apply final processing
        self.post_init()
 
    def get_input_embeddings(self):
        return self.embed_tokens
 
    def set_input_embeddings(self, value):
        self.embed_tokens = value
 
    # Copied from transformers.models.bart.modeling_bart.BartDecoder._prepare_decoder_attention_mask
    def _prepare_decoder_attention_mask(self, attention_mask, input_shape, inputs_embeds, past_key_values_length):
        # create causal mask
        # [bsz, seq_len] -> [bsz, 1, tgt_seq_len, src_seq_len]
        combined_attention_mask = None
        if input_shape[-1] > 1:
            combined_attention_mask = _make_causal_mask(
                input_shape,
                inputs_embeds.dtype,
                device=inputs_embeds.device,
                past_key_values_length=past_key_values_length,
            )
 
        if attention_mask is not None:
            # [bsz, seq_len] -> [bsz, 1, tgt_seq_len, src_seq_len]
            expanded_attn_mask = _expand_mask(attention_mask, inputs_embeds.dtype, tgt_len=input_shape[-1]).to(
                inputs_embeds.device
            )
            combined_attention_mask = (
                expanded_attn_mask if combined_attention_mask is None else expanded_attn_mask + combined_attention_mask
            )
 
        return combined_attention_mask
 
    @add_start_docstrings_to_model_forward(INTERNLM_INPUTS_DOCSTRING)
    def forward(
        self,
        input_ids: torch.LongTensor = None,
        attention_mask: Optional[torch.Tensor] = None,
        position_ids: Optional[torch.LongTensor] = None,
        past_key_values: Optional[List[torch.FloatTensor]] = None,
        inputs_embeds: Optional[torch.FloatTensor] = None,
        use_cache: Optional[bool] = None,
        output_attentions: Optional[bool] = None,
        output_hidden_states: Optional[bool] = None,
        return_dict: Optional[bool] = None,
    ) -> Union[Tuple, BaseModelOutputWithPast]:
        output_attentions = output_attentions if output_attentions is not None else self.config.output_attentions
        output_hidden_states = (
            output_hidden_states if output_hidden_states is not None else self.config.output_hidden_states
        )
        use_cache = use_cache if use_cache is not None else self.config.use_cache
 
        return_dict = return_dict if return_dict is not None else self.config.use_return_dict
 
        # retrieve input_ids and inputs_embeds
        if input_ids is not None and inputs_embeds is not None:
            raise ValueError("You cannot specify both decoder_input_ids and decoder_inputs_embeds at the same time")
        elif input_ids is not None:
            batch_size, seq_length = input_ids.shape
        elif inputs_embeds is not None:
            batch_size, seq_length, _ = inputs_embeds.shape
        else:
            raise ValueError("You have to specify either decoder_input_ids or decoder_inputs_embeds")
 
        seq_length_with_past = seq_length
        past_key_values_length = 0
 
        if past_key_values is not None:
            past_key_values_length = past_key_values[0][0].shape[2]
            seq_length_with_past = seq_length_with_past + past_key_values_length
 
        if position_ids is None:
            device = input_ids.device if input_ids is not None else inputs_embeds.device
            position_ids = torch.arange(
                past_key_values_length, seq_length + past_key_values_length, dtype=torch.long, device=device
            )
            position_ids = position_ids.unsqueeze(0).view(-1, seq_length)
        else:
            position_ids = position_ids.view(-1, seq_length).long()
 
        if inputs_embeds is None:
            inputs_embeds = self.embed_tokens(input_ids)
        # embed positions
        if attention_mask is None:
            attention_mask = torch.ones(
                (batch_size, seq_length_with_past), dtype=torch.bool, device=inputs_embeds.device
            )
        attention_mask = self._prepare_decoder_attention_mask(
            attention_mask, (batch_size, seq_length), inputs_embeds, past_key_values_length
        )
 
        hidden_states = inputs_embeds
 
        if self.gradient_checkpointing and self.training:
            if use_cache:
                logger.warning_once(
                    "`use_cache=True` is incompatible with gradient checkpointing. Setting `use_cache=False`..."
                )
                use_cache = False
 
        # decoder layers
        all_hidden_states = () if output_hidden_states else None
        all_self_attns = () if output_attentions else None
        next_decoder_cache = () if use_cache else None
 
        for idx, decoder_layer in enumerate(self.layers):
            if output_hidden_states:
                all_hidden_states += (hidden_states,)
 
            past_key_value = past_key_values[idx] if past_key_values is not None else None
 
            if self.gradient_checkpointing and self.training:
 
                def create_custom_forward(module):
                    def custom_forward(*inputs):
                        # None for past_key_value
                        return module(*inputs, output_attentions, None)
 
                    return custom_forward
 
                layer_outputs = torch.utils.checkpoint.checkpoint(
                    create_custom_forward(decoder_layer),
                    hidden_states,
                    attention_mask,
                    position_ids,
                    None,
                )
            else:
                layer_outputs = decoder_layer(
                    hidden_states,
                    attention_mask=attention_mask,
                    position_ids=position_ids,
                    past_key_value=past_key_value,
                    output_attentions=output_attentions,
                    use_cache=use_cache,
                )
 
            hidden_states = layer_outputs[0]
 
            if use_cache:
                next_decoder_cache += (layer_outputs[2 if output_attentions else 1],)
 
            if output_attentions:
                all_self_attns += (layer_outputs[1],)
 
        hidden_states = self.norm(hidden_states)
 
        # add hidden states from the last decoder layer
        if output_hidden_states:
            all_hidden_states += (hidden_states,)
 
        next_cache = next_decoder_cache if use_cache else None
        if not return_dict:
            return tuple(v for v in [hidden_states, next_cache, all_hidden_states, all_self_attns] if v is not None)
        return BaseModelOutputWithPast(
            last_hidden_state=hidden_states,
            past_key_values=next_cache,
            hidden_states=all_hidden_states,
            attentions=all_self_attns,
        )
 
 
class InternLMForCausalLM(InternLMPreTrainedModel):
    _auto_class = "AutoModelForCausalLM"
 
    def __init__(self, config):
        super().__init__(config)
        self.model = InternLMModel(config)
 
        self.lm_head = nn.Linear(config.hidden_size, config.vocab_size, bias=False)
 
        # Initialize weights and apply final processing
        self.post_init()
 
    def get_input_embeddings(self):
        return self.model.embed_tokens
 
    def set_input_embeddings(self, value):
        self.model.embed_tokens = value
 
    def get_output_embeddings(self):
        return self.lm_head
 
    def set_output_embeddings(self, new_embeddings):
        self.lm_head = new_embeddings
 
    def set_decoder(self, decoder):
        self.model = decoder
 
    def get_decoder(self):
        return self.model
 
    @add_start_docstrings_to_model_forward(INTERNLM_INPUTS_DOCSTRING)
    @replace_return_docstrings(output_type=CausalLMOutputWithPast, config_class=_CONFIG_FOR_DOC)
    def forward(
        self,
        input_ids: torch.LongTensor = None,
        attention_mask: Optional[torch.Tensor] = None,
        position_ids: Optional[torch.LongTensor] = None,
        past_key_values: Optional[List[torch.FloatTensor]] = None,
        inputs_embeds: Optional[torch.FloatTensor] = None,
        labels: Optional[torch.LongTensor] = None,
        use_cache: Optional[bool] = None,
        output_attentions: Optional[bool] = None,
        output_hidden_states: Optional[bool] = None,
        return_dict: Optional[bool] = None,
    ) -> Union[Tuple, CausalLMOutputWithPast]:
        r"""
        Args:
            labels (`torch.LongTensor` of shape `(batch_size, sequence_length)`, *optional*):
                Labels for computing the masked language modeling loss. Indices should either be in `[0, ...,
                config.vocab_size]` or -100 (see `input_ids` docstring). Tokens with indices set to `-100` are ignored
                (masked), the loss is only computed for the tokens with labels in `[0, ..., config.vocab_size]`.
        Returns:
        Example:
        ```python
        >>> from transformers import AutoTokenizer, InternLMForCausalLM
        >>> model = InternLMForCausalLM.from_pretrained(PATH_TO_CONVERTED_WEIGHTS)
        >>> tokenizer = AutoTokenizer.from_pretrained(PATH_TO_CONVERTED_TOKENIZER)
        >>> prompt = "Hey, are you consciours? Can you talk to me?"
        >>> inputs = tokenizer(prompt, return_tensors="pt")
        >>> # Generate
        >>> generate_ids = model.generate(inputs.input_ids, max_length=30)
        >>> tokenizer.batch_decode(generate_ids, skip_special_tokens=True, clean_up_tokenization_spaces=False)[0]
        "Hey, are you consciours? Can you talk to me?\nI'm not consciours, but I can talk to you."
        ```"""
 
        output_attentions = output_attentions if output_attentions is not None else self.config.output_attentions
        output_hidden_states = (
            output_hidden_states if output_hidden_states is not None else self.config.output_hidden_states
        )
        return_dict = return_dict if return_dict is not None else self.config.use_return_dict
 
        # decoder outputs consists of (dec_features, layer_state, dec_hidden, dec_attn)
        outputs = self.model(
            input_ids=input_ids,
            attention_mask=attention_mask,
            position_ids=position_ids,
            past_key_values=past_key_values,
            inputs_embeds=inputs_embeds,
            use_cache=use_cache,
            output_attentions=output_attentions,
            output_hidden_states=output_hidden_states,
            return_dict=return_dict,
        )
 
        hidden_states = outputs[0]
        logits = self.lm_head(hidden_states)
 
        loss = None
        if labels is not None:
            # Shift so that tokens < n predict n
            shift_logits = logits[..., :-1, :].contiguous()
            shift_labels = labels[..., 1:].contiguous()
            # Flatten the tokens
            loss_fct = CrossEntropyLoss()
            shift_logits = shift_logits.view(-1, self.config.vocab_size)
            shift_labels = shift_labels.view(-1)
            # Enable model parallelism
            shift_labels = shift_labels.to(shift_logits.device)
            loss = loss_fct(shift_logits, shift_labels)
 
        if not return_dict:
            output = (logits,) + outputs[1:]
            return (loss,) + output if loss is not None else output
 
        return CausalLMOutputWithPast(
            loss=loss,
            logits=logits,
            past_key_values=outputs.past_key_values,
            hidden_states=outputs.hidden_states,
            attentions=outputs.attentions,
        )
 
    def prepare_inputs_for_generation(
        self, input_ids, past_key_values=None, attention_mask=None, inputs_embeds=None, **kwargs
    ):
        if past_key_values:
            input_ids = input_ids[:, -1:]
 
        position_ids = kwargs.get("position_ids", None)
        if attention_mask is not None and position_ids is None:
            # create position_ids on the fly for batch generation
            position_ids = attention_mask.long().cumsum(-1) - 1
            position_ids.masked_fill_(attention_mask == 0, 1)
            if past_key_values:
                position_ids = position_ids[:, -1].unsqueeze(-1)
 
        # if `inputs_embeds` are passed, we only want to use them in the 1st generation step
        if inputs_embeds is not None and past_key_values is None:
            model_inputs = {"inputs_embeds": inputs_embeds}
        else:
            model_inputs = {"input_ids": input_ids}
 
        model_inputs.update(
            {
                "position_ids": position_ids,
                "past_key_values": past_key_values,
                "use_cache": kwargs.get("use_cache"),
                "attention_mask": attention_mask,
            }
        )
        return model_inputs
 
    @staticmethod
    def _reorder_cache(past_key_values, beam_idx):
        reordered_past = ()
        for layer_past in past_key_values:
            reordered_past += (tuple(past_state.index_select(0, beam_idx) for past_state in layer_past),)
        return reordered_past
    
    def build_inputs(self, tokenizer, query: str, history: List[Tuple[str, str]] = []):
        prompt = ""
        for record in history:
            prompt += f"""<|User|>:{record[0]}\n<|Bot|>:{record[1]}\n"""
        if len(prompt) == 0:
            prompt += ""
        prompt += f"""<|User|>:{query}\n<|Bot|>:"""
        return tokenizer([prompt], return_tensors="pt")
    
    @torch.no_grad()
    def chat(self, 
             tokenizer, 
             query: str,
             history: List[Tuple[str, str]] = [], 
             streamer: Optional[BaseStreamer] = None,
             max_new_tokens: int = 1024,
             do_sample: bool = True,
             temperature: float = 0.8,
             top_p: float = 0.8,
             **kwargs):
        inputs = self.build_inputs(tokenizer, query, history)
        inputs = {k: v.to(self.device) for k, v in inputs.items() if torch.is_tensor(v)}
        outputs = self.generate(**inputs, 
                                streamer=streamer, 
                                max_new_tokens=max_new_tokens, 
                                do_sample=do_sample, 
                                temperature=temperature, 
                                top_p=top_p, 
                                **kwargs)
        outputs = outputs[0].cpu().tolist()[len(inputs["input_ids"][0]):]
        response = tokenizer.decode(outputs, skip_special_tokens=True)
        response = response.split("")[0]
        history = history + [(query, response)]
        return response, history
    
    @torch.no_grad()
    def stream_chat(self, 
                    tokenizer,
                    query: str,
                    history: List[Tuple[str, str]] = [], 
                    max_new_tokens: int = 1024,
                    do_sample: bool = True,
                    temperature: float = 0.8,
                    top_p: float = 0.8,
                    **kwargs):
        """
        Return a generator in format: (response, history)
        Eg.
        ('你好，有什么可以帮助您的吗', [('你好', '你好，有什么可以帮助您的吗')])
        ('你好，有什么可以帮助您的吗？', [('你好', '你好，有什么可以帮助您的吗？')])
        """
 
        response_queue = queue.Queue(maxsize=20)
 
        class ChatStreamer(BaseStreamer):
            def __init__(self, tokenizer) -> None:
                super().__init__()
                self.tokenizer = tokenizer
                self.queue = response_queue
                self.query = query
                self.history = history
                self.response = ""
                self.received_inputs = False
                self.queue.put((self.response, history + [(self.query, self.response)]))
 
            def put(self, value):
                if len(value.shape) > 1 and value.shape[0] > 1:
                    raise ValueError("ChatStreamer only supports batch size 1")
                elif len(value.shape) > 1:
                    value = value[0]
 
                if not self.received_inputs:
                    # The first received value is input_ids, ignore here
                    self.received_inputs = True
                    return
 
                token = self.tokenizer.decode([value[-1]], skip_special_tokens=True)
                if token.strip() != "":
                    self.response = self.response + token
                    history = self.history + [(self.query, self.response)]
                    self.queue.put((self.response, history))
 
            def end(self):
                self.queue.put(None)
 
        def stream_producer():
            return self.chat(
                tokenizer=tokenizer,
                query=query,
                streamer=ChatStreamer(tokenizer=tokenizer),
                history=history, 
                max_new_tokens=max_new_tokens,
                do_sample=do_sample,
                temperature=temperature,
                top_p=top_p,
                **kwargs
            )
 
        def consumer():
            producer = threading.Thread(target=stream_producer)
            producer.start()
            while True:
                res = response_queue.get()
                if res is None:
                    return
                yield res
 
        return consumer()
 
 
@add_start_docstrings(
    """
    The InternLM Model transformer with a sequence classification head on top (linear layer).
    [`InternLMForSequenceClassification`] uses the last token in order to do the classification, as other causal models
    (e.g. GPT-2) do.
    Since it does classification on the last token, it requires to know the position of the last token. If a
    `pad_token_id` is defined in the configuration, it finds the last token that is not a padding token in each row. If
    no `pad_token_id` is defined, it simply takes the last value in each row of the batch. Since it cannot guess the
    padding tokens when `inputs_embeds` are passed instead of `input_ids`, it does the same (take the last value in
    each row of the batch).
    """,
    INTERNLM_START_DOCSTRING,
)
class InternLMForSequenceClassification(InternLMPreTrainedModel):
    _keys_to_ignore_on_load_missing = [r"lm_head.weight"]
 
    def __init__(self, config):
        super().__init__(config)
        self.num_labels = config.num_labels
        self.model = InternLMModel(config)
        self.score = nn.Linear(config.hidden_size, self.num_labels, bias=False)
 
        # Initialize weights and apply final processing
        self.post_init()
 
    def get_input_embeddings(self):
        return self.model.embed_tokens
 
    def set_input_embeddings(self, value):
        self.model.embed_tokens = value
 
    @add_start_docstrings_to_model_forward(INTERNLM_INPUTS_DOCSTRING)
    def forward(
        self,
        input_ids: torch.LongTensor = None,
        attention_mask: Optional[torch.Tensor] = None,
        position_ids: Optional[torch.LongTensor] = None,
        past_key_values: Optional[List[torch.FloatTensor]] = None,
        inputs_embeds: Optional[torch.FloatTensor] = None,
        labels: Optional[torch.LongTensor] = None,
        use_cache: Optional[bool] = None,
        output_attentions: Optional[bool] = None,
        output_hidden_states: Optional[bool] = None,
        return_dict: Optional[bool] = None,
    ) -> Union[Tuple, SequenceClassifierOutputWithPast]:
        r"""
        labels (`torch.LongTensor` of shape `(batch_size,)`, *optional*):
            Labels for computing the sequence classification/regression loss. Indices should be in `[0, ...,
            config.num_labels - 1]`. If `config.num_labels == 1` a regression loss is computed (Mean-Square loss), If
            `config.num_labels > 1` a classification loss is computed (Cross-Entropy).
        """
        return_dict = return_dict if return_dict is not None else self.config.use_return_dict
 
        transformer_outputs = self.model(
            input_ids,
            attention_mask=attention_mask,
            position_ids=position_ids,
            past_key_values=past_key_values,
            inputs_embeds=inputs_embeds,
            use_cache=use_cache,
            output_attentions=output_attentions,
            output_hidden_states=output_hidden_states,
            return_dict=return_dict,
        )
        hidden_states = transformer_outputs[0]
        logits = self.score(hidden_states)
 
        if input_ids is not None:
            batch_size = input_ids.shape[0]
        else:
            batch_size = inputs_embeds.shape[0]
 
        if self.config.pad_token_id is None and batch_size != 1:
            raise ValueError("Cannot handle batch sizes > 1 if no padding token is defined.")
        if self.config.pad_token_id is None:
            sequence_lengths = -1
        else:
            if input_ids is not None:
                sequence_lengths = (torch.ne(input_ids, self.config.pad_token_id).sum(-1) - 1).to(logits.device)
            else:
                sequence_lengths = -1
 
        pooled_logits = logits[torch.arange(batch_size, device=logits.device), sequence_lengths]
 
        loss = None
        if labels is not None:
            labels = labels.to(logits.device)
            if self.config.problem_type is None:
                if self.num_labels == 1:
                    self.config.problem_type = "regression"
                elif self.num_labels > 1 and (labels.dtype == torch.long or labels.dtype == torch.int):
                    self.config.problem_type = "single_label_classification"
                else:
                    self.config.problem_type = "multi_label_classification"
 
            if self.config.problem_type == "regression":
                loss_fct = MSELoss()
                if self.num_labels == 1:
                    loss = loss_fct(pooled_logits.squeeze(), labels.squeeze())
                else:
                    loss = loss_fct(pooled_logits, labels)
            elif self.config.problem_type == "single_label_classification":
                loss_fct = CrossEntropyLoss()
                loss = loss_fct(pooled_logits.view(-1, self.num_labels), labels.view(-1))
            elif self.config.problem_type == "multi_label_classification":
                loss_fct = BCEWithLogitsLoss()
                loss = loss_fct(pooled_logits, labels)
        if not return_dict:
            output = (pooled_logits,) + transformer_outputs[1:]
            return ((loss,) + output) if loss is not None else output
 
        return SequenceClassifierOutputWithPast(
            loss=loss,
            logits=pooled_logits,
            past_key_values=transformer_outputs.past_key_values,
            hidden_states=transformer_outputs.hidden_states,
            attentions=transformer_outputs.attentions,
        )

    dim = 8
    base = 10000
    device = None
    # 根据概率计算θ, dim/2个θ
    inv_freq = 1.0 / (base ** (torch.arange(0, dim, 2).float().to(device) / dim))

tensor([1.0000, 0.1000, 0.0100, 0.0010])

    inv_freq_cached = inv_freq
    max_seq_len_cached = max_position_embeddings
    t = torch.arange(max_seq_len_cached, device=device, dtype=inv_freq.dtype)
    freqs = torch.einsum("i,j->ij", t, inv_freq)

tensor([0.0000e+00, 1.0000e+00, 2.0000e+00,  ..., 4.0660e+03, 4.0670e+03, 4.0680e+03])

emb = torch.cat((freqs, freqs), dim=-1)
self.register_buffer("cos_cached", emb.cos()[None, None, :, :], persistent=False)
self.register_buffer("sin_cached", emb.sin()[None, None, :, :], persistent=False)

[θ1,θ2,θ3,θ4] [4069, 4] x 2 -> [θ1,θ2,θ3,θ4,θ1,θ2,θ3,θ4] torch.Size([4069, 8])

# This `if` block is unlikely to be run after we build sin/cos in `__init__`. Keep the logic here just in case.

torch.Size([1, 1, 4069, 8])

v1: [q0,q1,q2,q3,q4,q5,q6,q7] -> [-q4,-q5,-q6,-q7,q0,q1,q2,q3]

    # v1: [q0,q1,q2,q3,q4,q5,q6,q7] -> [-q4,-q5,-q6,-q7,q0,q1,q2,q3]
 
    # [cosθ1, cosθ2, cosθ3, cosθ4, cosθ1, cosθ2, cosθ3, cosθ4]
    # [sinθ1, sinθ2, sinθ3, sinθ4, sinθ1, sinθ2, sinθ3, sinθ4]

def rotate_half_v2(x):
    x1 = x[..., ::2]
    x2 = x[..., 1::2]
    return torch.cat((-x2, x1), dim=-1)

    # v2: [q0,q1,q2,q3,q4,q5,q6,q7] -> [-q1,-q3,-q5,-q7,q0,q2,q4,q6]
 
    # [cosθ1, cosθ2, cosθ3, cosθ4, cosθ1, cosθ2, cosθ3, cosθ4]
    # [sinθ1, sinθ2, sinθ3, sinθ4, sinθ1, sinθ2, sinθ3, sinθ4]

    # Query、Key、Value 信息
    batch_size = 2
    query_length = 10
    num_heads = 4
    head_dim = 8
    past_key_value = None
 
    # 获取 Query、Key、Value Matrix
    query_states = torch.rand(batch_size, query_length, num_heads, head_dim).transpose(1, 2)
    key_states = torch.rand(batch_size, query_length, num_heads, head_dim).transpose(1, 2)
    value_states = torch.rand(batch_size, query_length, num_heads, head_dim).transpose(1, 2)

tensor([[[[2.9183e-01, 5.9775e-02, 3.5425e-01, 2.7915e-01, 5.2601e-01,
           8.3887e-01, 3.3162e-01, 1.3850e-01],
          [2.8997e-01, 1.5348e-01, 6.1809e-01, 6.7418e-01, 2.4923e-02,
           3.9746e-01, 7.3853e-01, 3.4693e-02],
          [4.1272e-01, 8.2844e-01, 3.2101e-01, 9.3669e-01, 9.9066e-01,
           7.3192e-01, 4.9623e-01, 3.6537e-01],
          [9.3123e-01, 6.0658e-01, 5.1689e-02, 4.8995e-01, 8.3494e-01,
           9.5322e-01, 4.4755e-01, 4.6204e-01],
          [2.2266e-01, 7.8899e-01, 3.1150e-01, 8.6672e-01, 5.7448e-01,
           1.8205e-02, 1.4669e-01, 1.4783e-01],
          [2.3940e-01, 8.3407e-01, 2.4727e-01, 4.8705e-02, 7.4942e-01,
           3.3560e-01, 5.5196e-01, 9.0414e-01],
          [5.7593e-01, 3.1633e-01, 1.8330e-01, 7.7621e-01, 5.8697e-01,
           1.8219e-01, 8.6412e-01, 4.4979e-02],
          [7.9487e-01, 2.2888e-02, 7.1163e-01, 1.7849e-01, 5.7600e-01,
           8.9930e-01, 3.5231e-01, 1.5648e-01],
          [5.2896e-02, 3.2068e-01, 7.3980e-01, 7.7020e-01, 4.4519e-01,
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          [6.8193e-01, 4.9851e-01, 1.0965e-02, 1.3012e-02, 8.7816e-01,
           8.8564e-01, 5.6333e-01, 8.6411e-01]],
 
         [[6.8500e-01, 7.0738e-01, 2.9308e-01, 9.2951e-01, 6.1441e-02,
           3.6203e-02, 6.3757e-01, 4.7221e-01],
          [4.3020e-01, 5.7936e-01, 5.8797e-01, 5.2842e-01, 2.1757e-01,
           8.2139e-01, 8.1286e-01, 5.1831e-01],
          [5.0025e-01, 4.9856e-01, 8.6077e-01, 4.8580e-01, 6.7289e-01,
           1.6959e-01, 9.1899e-01, 7.0452e-02],
          [6.8620e-01, 5.9861e-01, 6.1024e-01, 9.4969e-01, 8.7641e-01,
           2.1290e-01, 2.9642e-01, 6.4476e-01],
          [4.0705e-02, 3.1697e-02, 9.2447e-01, 6.6854e-02, 2.7703e-01,
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          [7.6717e-02, 3.4182e-01, 8.6993e-01, 3.9317e-01, 5.4155e-01,
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          [7.2185e-01, 2.5139e-01, 4.0992e-01, 2.3913e-01, 9.5667e-01,
           8.7461e-01, 5.1859e-02, 6.5691e-01],
          [5.9603e-01, 5.5759e-01, 6.1473e-01, 4.7875e-01, 7.2805e-01,
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          [8.4941e-01, 5.4692e-01, 1.9650e-01, 7.9873e-01, 4.7650e-01,
           4.1342e-01, 5.8423e-01, 4.6351e-02]],
 
         [[6.4038e-01, 9.8298e-01, 3.1733e-01, 9.0533e-01, 5.8110e-01,
           4.9543e-01, 2.5562e-01, 9.5430e-01],
          [8.4416e-01, 1.5592e-01, 5.0446e-01, 9.4544e-01, 7.0241e-01,
           1.1321e-01, 9.0690e-01, 5.6367e-01],
          [6.8041e-01, 4.5696e-01, 8.0091e-01, 6.6836e-01, 3.1932e-01,
           2.5314e-02, 4.5553e-01, 3.2404e-03],
          [8.5800e-01, 6.9543e-01, 7.2294e-01, 1.3468e-01, 2.9467e-01,
           4.5119e-01, 6.0176e-01, 5.9708e-01],
          [4.7581e-01, 9.5652e-01, 3.6950e-01, 8.5315e-01, 1.5493e-01,
           4.5351e-01, 5.5274e-02, 3.9174e-01],
          [6.7330e-01, 4.2024e-01, 4.5872e-01, 3.6176e-01, 5.6271e-01,
           7.4545e-01, 1.9587e-01, 9.3089e-01],
          [3.6712e-01, 7.1372e-01, 1.8244e-01, 9.1078e-01, 1.4220e-01,
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          [6.0856e-01, 5.5768e-01, 4.7132e-01, 5.1918e-01, 8.2279e-01,
           5.6378e-01, 6.5475e-01, 8.8963e-01],
          [4.7355e-01, 8.9392e-01, 5.2195e-01, 9.9627e-01, 7.1131e-01,
           9.4066e-01, 9.6293e-01, 2.1514e-01],
          [6.6799e-01, 1.5106e-03, 1.0987e-01, 7.2399e-01, 5.0414e-01,
           2.2703e-01, 2.2963e-01, 3.0123e-01]],
 
         [[3.4204e-01, 5.2402e-01, 5.7015e-01, 2.3358e-02, 8.8628e-01,
           7.8516e-01, 3.6371e-01, 8.4997e-01],
          [1.8298e-01, 3.5044e-01, 2.1751e-01, 2.5430e-01, 1.2984e-01,
           1.5158e-01, 4.8908e-01, 9.8046e-01],
          [4.0914e-01, 9.2517e-01, 6.0098e-02, 4.6138e-01, 3.2282e-01,
           3.6191e-01, 3.0836e-01, 2.8638e-01],
          [3.8517e-01, 7.4692e-01, 8.9863e-01, 2.7790e-02, 3.9417e-01,
           9.2557e-01, 9.2542e-01, 7.1499e-01],
          [8.0929e-01, 8.2430e-01, 7.7641e-01, 4.4017e-01, 9.4351e-01,
           8.4430e-01, 3.9162e-01, 4.7323e-01],
          [4.4801e-02, 1.0588e-01, 2.5283e-01, 2.9120e-01, 8.0959e-01,
           6.7210e-01, 6.3844e-01, 2.8742e-01],
          [7.6291e-01, 7.9657e-01, 3.8156e-01, 8.2935e-01, 2.1237e-01,
           6.5650e-01, 7.8452e-01, 2.8097e-01],
          [2.0586e-01, 7.0199e-01, 5.1598e-01, 1.3107e-01, 7.8793e-01,
           1.1133e-01, 4.1328e-01, 1.5066e-01],
          [2.1346e-01, 4.1860e-01, 2.8186e-01, 6.9559e-03, 3.1852e-01,
           7.0937e-01, 8.8809e-01, 7.5510e-01],
          [9.1850e-01, 6.2456e-01, 9.4854e-01, 2.2026e-01, 8.4640e-01,
           4.0601e-02, 7.9473e-01, 8.2286e-01]]],
 
 
        [[[2.8326e-01, 9.0059e-01, 9.3148e-01, 4.2370e-01, 2.5814e-01,
           6.6909e-01, 1.0928e-02, 3.9296e-01],
          [5.1192e-01, 7.1407e-01, 8.8236e-01, 3.7678e-01, 2.2305e-01,
           3.1705e-01, 6.3382e-01, 7.9083e-01],
          [8.0426e-01, 5.3209e-01, 2.9103e-01, 1.0520e-02, 6.4828e-01,
           7.5650e-01, 4.3391e-01, 1.9701e-01],
          [6.4033e-01, 9.4359e-01, 2.6629e-01, 5.5410e-01, 8.4910e-01,
           7.3342e-01, 1.4064e-01, 8.7489e-01],
          [1.7254e-01, 9.6152e-01, 3.8131e-01, 3.8041e-01, 6.3002e-01,
           7.7604e-01, 2.7887e-01, 6.7200e-01],
          [2.0756e-01, 7.8653e-01, 4.6171e-01, 4.1734e-01, 5.7438e-01,
           3.6887e-01, 2.0821e-01, 9.6894e-01],
          [7.8206e-01, 3.9417e-01, 1.6260e-01, 7.2534e-01, 7.5993e-02,
           9.3412e-01, 7.7624e-02, 9.3928e-01],
          [9.7513e-02, 6.1370e-01, 5.7725e-01, 3.4250e-01, 1.1776e-01,
           5.8991e-01, 3.6010e-01, 1.5810e-01],
          [9.6534e-03, 3.0747e-01, 5.6363e-01, 2.5881e-01, 3.7878e-01,
           4.2585e-01, 4.9041e-01, 7.6013e-01],
          [4.1924e-01, 1.4514e-01, 6.4527e-01, 9.7834e-01, 6.0451e-01,
           7.6843e-01, 4.1792e-01, 8.4611e-01]],
 
         [[2.4590e-01, 3.4102e-01, 6.0761e-01, 3.9460e-01, 4.5805e-01,
           7.6451e-01, 2.3557e-03, 5.7520e-01],
          [2.5920e-01, 9.7934e-01, 3.2950e-01, 7.1681e-01, 2.3382e-01,
           1.1517e-01, 6.4881e-01, 9.3044e-01],
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          [5.5971e-01, 7.1264e-01, 5.3499e-01, 2.8207e-02, 6.8861e-01,
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           4.8248e-02, 4.4249e-01, 6.4919e-01],
          [4.0007e-01, 6.7373e-01, 8.6021e-01, 3.7193e-01, 3.9114e-01,
           6.1232e-01, 3.7100e-01, 8.6428e-01],
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           4.0885e-01, 8.1395e-01, 9.3831e-01],
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           9.4150e-01, 6.7335e-01, 4.2026e-02],
          [6.9338e-01, 5.0140e-01, 7.2334e-01, 3.3677e-01, 7.4549e-01,
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           7.9191e-01, 6.8408e-01, 6.8683e-01]],
 
         [[5.0144e-02, 7.6259e-01, 6.7326e-01, 9.8079e-01, 1.1124e-01,
           4.2291e-01, 1.8365e-02, 4.4412e-01],
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           3.1403e-01, 7.0689e-01, 1.6075e-02],
          [3.3388e-01, 7.9689e-01, 1.0852e-02, 9.4362e-01, 7.6169e-01,
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          [9.1527e-01, 8.0093e-01, 1.0186e-01, 4.6437e-01, 3.7368e-01,
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          [5.9391e-01, 6.7338e-01, 9.4407e-01, 5.5762e-01, 5.8258e-01,
           1.8335e-02, 4.0466e-01, 1.5728e-01],
          [5.5420e-02, 8.7254e-01, 4.2972e-01, 7.3742e-01, 4.0287e-01,
           9.8190e-01, 2.8029e-01, 4.3533e-01],
          [3.2755e-01, 6.7667e-01, 5.8706e-01, 2.0986e-01, 9.5659e-01,
           7.9053e-01, 7.8864e-01, 9.1482e-02],
          [5.7617e-01, 7.2545e-02, 9.7183e-01, 6.0007e-01, 6.8718e-01,
           9.3688e-01, 7.3142e-01, 3.5678e-01],
          [7.1092e-01, 9.9054e-01, 6.8759e-01, 9.4545e-04, 7.7356e-01,
           3.8864e-01, 3.4851e-01, 7.0661e-01]],
 
         [[8.3643e-01, 5.9542e-01, 2.8547e-01, 4.0909e-01, 4.0864e-01,
           6.4102e-01, 6.2927e-01, 7.0807e-01],
          [2.2826e-01, 3.7989e-01, 4.4519e-01, 9.7904e-01, 2.2382e-01,
           5.0949e-01, 6.7909e-01, 4.7000e-01],
          [6.3309e-01, 5.1963e-01, 3.8026e-01, 6.9920e-01, 3.8465e-01,
           2.5132e-01, 4.9886e-01, 4.6673e-01],
          [9.5898e-01, 8.0877e-01, 6.4463e-01, 7.9780e-01, 7.6710e-01,
           1.4926e-01, 8.9849e-01, 4.8631e-01],
          [7.8407e-01, 1.0189e-01, 9.9365e-01, 9.5311e-01, 4.6489e-01,
           6.7042e-01, 1.6722e-01, 7.3440e-01],
          [5.4366e-01, 4.0804e-01, 2.1323e-01, 1.5707e-01, 3.4207e-01,
           2.5938e-01, 6.3546e-01, 7.6161e-01],
          [8.2502e-01, 6.5296e-01, 5.5894e-01, 6.8899e-01, 7.9576e-01,
           2.8821e-01, 5.0587e-01, 7.1531e-01],
          [7.5482e-02, 7.7817e-01, 8.5415e-01, 8.1673e-01, 7.6846e-01,
           6.3890e-01, 3.5124e-01, 9.1766e-01],
          [6.8266e-01, 8.7167e-01, 1.1057e-01, 5.7511e-01, 1.4933e-01,
           1.3559e-01, 1.2099e-01, 5.2032e-01],
          [7.0112e-01, 1.2467e-01, 3.8753e-01, 8.5428e-01, 2.8684e-01,
           6.2195e-01, 6.6607e-01, 1.9613e-01]]]])