
from __future__ import annotations

from collections.abc import Callable

UpperCamelCase_   	=	list[list[float | int]]






def 		_UpperCAmelCase   (	UpperCamelCase:     Matrix			,							UpperCamelCase:     Matrix  ):






	"""simple docstring"""


	__lowerCAmelCase         =							len(UpperCamelCase  )
	__lowerCAmelCase         =							[[0 for _ in range(size + 1  )] for _ in range(UpperCamelCase  )]
	__lowerCAmelCase         =							42
	__lowerCAmelCase         =							42
	__lowerCAmelCase         =							42
	__lowerCAmelCase         =							42
	__lowerCAmelCase         =							42
	__lowerCAmelCase         =							42

	for row in range(UpperCamelCase  ):
		for col in range(UpperCamelCase  ):
			__lowerCAmelCase         =							matrix[row][col]

		__lowerCAmelCase         =							vector[row][0]

	__lowerCAmelCase         =							0
	__lowerCAmelCase         =							0
	while row < size and col < size:
		# pivoting
		__lowerCAmelCase         =							max((abs(augmented[rowa][col]  ), rowa) for rowa in range(UpperCamelCase			,							UpperCamelCase  )  )[
		    1
		]
		if augmented[pivot_row][col] == 0:
			col += 1
			continue
		else:
			__lowerCAmelCase   ,						__lowerCAmelCase         =							augmented[pivot_row], augmented[row]

		for rowa in range(row + 1			,							UpperCamelCase  ):
			__lowerCAmelCase         =							augmented[rowa][col] / augmented[row][col]
			__lowerCAmelCase         =							0
			for cola in range(col + 1			,							size + 1  ):
				augmented[rowa][cola] -= augmented[row][cola] * ratio

		row += 1
		col += 1

	# back substitution
	for col in range(1			,							UpperCamelCase  ):
		for row in range(UpperCamelCase  ):
			__lowerCAmelCase         =							augmented[row][col] / augmented[col][col]
			for cola in range(UpperCamelCase			,							size + 1  ):
				augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
	return [
	    [round(augmented[row][size] / augmented[row][row]			,							1_0  )] for row in range(UpperCamelCase  )
	]






def 		_UpperCAmelCase   (	UpperCamelCase:     list[int]  ):






	"""simple docstring"""


	__lowerCAmelCase         =							len(UpperCamelCase  )
	__lowerCAmelCase         =							[[0 for _ in range(UpperCamelCase  )] for _ in range(UpperCamelCase  )]
	__lowerCAmelCase         =							[[0] for _ in range(UpperCamelCase  )]
	__lowerCAmelCase         =							42
	__lowerCAmelCase         =							42
	__lowerCAmelCase         =							42
	__lowerCAmelCase         =							42

	for x_val, y_val in enumerate(UpperCamelCase  ):
		for col in range(UpperCamelCase  ):
			__lowerCAmelCase         =							(x_val + 1) ** (size - col - 1)
		__lowerCAmelCase         =							y_val

	__lowerCAmelCase         =							solve(UpperCamelCase			,							UpperCamelCase  )

	def interpolated_func(UpperCamelCase:     int  ) -> int:
		return sum(
		    round(coeffs[x_val][0]  ) * (var ** (size - x_val - 1))
		    for x_val in range(UpperCamelCase  )  )

	return interpolated_func






def 		_UpperCAmelCase   (	UpperCamelCase:     int  ):






	"""simple docstring"""


	return (
	    1
	    - variable
	    + variable**2
	    - variable**3
	    + variable**4
	    - variable**5
	    + variable**6
	    - variable**7
	    + variable**8
	    - variable**9
	    + variable**1_0
	)






def 		_UpperCAmelCase   (	UpperCamelCase:     Callable[[int], int] = question_function			,							UpperCamelCase:     int = 1_0  ):






	"""simple docstring"""


	__lowerCAmelCase         =							[func(UpperCamelCase  ) for x_val in range(1			,							order + 1  )]

	__lowerCAmelCase         =							[
	    interpolate(data_points[:max_coeff]  ) for max_coeff in range(1			,							order + 1  )
	]

	__lowerCAmelCase         =							0
	__lowerCAmelCase         =							42
	__lowerCAmelCase         =							42

	for poly in polynomials:
		__lowerCAmelCase         =							1
		while func(UpperCamelCase  ) == poly(UpperCamelCase  ):
			x_val += 1

		ret += poly(UpperCamelCase  )

	return ret


if __name__ == "__main__":
					print(f'''{solution() = }''')








import json
import os
from dataclasses import dataclass
from functools import partial
from typing import Callable

import flax.linen as nn
import jax
import jax.numpy as jnp
import joblib
import optax
import wandb
from flax import jax_utils, struct, traverse_util
from flax.serialization import from_bytes, to_bytes
from flax.training import train_state
from flax.training.common_utils import shard
from tqdm.auto import tqdm

from transformers import BigBirdConfig, FlaxBigBirdForQuestionAnswering
from transformers.models.big_bird.modeling_flax_big_bird import FlaxBigBirdForQuestionAnsweringModule

class       a		(   __UpperCAmelCase					):
							lowercase_		:						BigBirdConfig
							lowercase_		:						jnp.dtype   							=  jnp.floataa
							lowercase_		:						bool   							=  True







							def       UpperCAmelCase__					(						self  :  Optional[int] ):
								"""simple docstring"""







								super().setup()
								__lowerCAmelCase         =							nn.Dense(5      ,				dtype=self.dtype )







							def __call__(						self  :  Optional[Any]      ,				*snake_case__  :  List[str]      ,				**snake_case__  :  str ):
								"""simple docstring"""







								__lowerCAmelCase         =							super().__call__(*snake_case__      ,				**snake_case__ )
								__lowerCAmelCase         =							self.cls(outputs[2] )
								return outputs[:2] + (cls_out,)

class       a		(   __UpperCAmelCase					):
							lowercase_		:						List[str]   							=  FlaxBigBirdForNaturalQuestionsModule






def 		_UpperCAmelCase   (	UpperCamelCase:     Optional[Any]			,							UpperCamelCase:     List[str]			,							UpperCamelCase:     Optional[Any]			,							UpperCamelCase:     List[Any]			,							UpperCamelCase:     Optional[Any]			,							UpperCamelCase:     int  ):






	"""simple docstring"""


	def cross_entropy(UpperCamelCase:     Union[str, Any]			,							UpperCamelCase:     List[Any]			,							UpperCamelCase:     Optional[int]=None  ):
		__lowerCAmelCase         =							logits.shape[-1]
		__lowerCAmelCase         =							(labels[..., None] == jnp.arange(UpperCamelCase  )[None]).astype("f4"  )
		__lowerCAmelCase         =							jax.nn.log_softmax(UpperCamelCase			,							axis=-1  )
		__lowerCAmelCase         =							-jnp.sum(labels * logits			,							axis=-1  )
		if reduction is not None:
			__lowerCAmelCase         =							reduction(UpperCamelCase  )
		return loss

	__lowerCAmelCase         =							partial(UpperCamelCase			,							reduction=jnp.mean  )
	__lowerCAmelCase         =							cross_entropy(UpperCamelCase			,							UpperCamelCase  )
	__lowerCAmelCase         =							cross_entropy(UpperCamelCase			,							UpperCamelCase  )
	__lowerCAmelCase         =							cross_entropy(UpperCamelCase			,							UpperCamelCase  )
	return (start_loss + end_loss + pooled_loss) / 3

@dataclass
class       a		:
							lowercase_		:						str   							=  "google/bigbird-roberta-base"
							lowercase_		:						int   							=  3_000
							lowercase_		:						int   							=  10_500

							lowercase_		:						int   							=  128
							lowercase_		:						int   							=  3

							lowercase_		:						int   							=  1
							lowercase_		:						int   							=  5

							# tx_args
							lowercase_		:						float   							=  3e-5
							lowercase_		:						float   							=  0.0
							lowercase_		:						int   							=  20_000
							lowercase_		:						float   							=  0.0095

							lowercase_		:						str   							=  "bigbird-roberta-natural-questions"
							lowercase_		:						str   							=  "training-expt"
							lowercase_		:						str   							=  "data/nq-training.jsonl"
							lowercase_		:						str   							=  "data/nq-validation.jsonl"







							def       UpperCAmelCase__					(						self  :  Optional[int] ):
								"""simple docstring"""







								os.makedirs(self.base_dir      ,				exist_ok=snake_case__ )
								__lowerCAmelCase         =							os.path.join(self.base_dir      ,				self.save_dir )
								__lowerCAmelCase         =							self.batch_size_per_device * jax.device_count()

@dataclass
class       a		:
							lowercase_		:						int
							lowercase_		:						int   							=  4_096  # no dynamic padding on TPUs







							def __call__(						self  :  List[Any]      ,				snake_case__  :  Union[str, Any] ):
								"""simple docstring"""







								__lowerCAmelCase         =							self.collate_fn(snake_case__ )
								__lowerCAmelCase         =							jax.tree_util.tree_map(snake_case__      ,				snake_case__ )
								return batch







							def       UpperCAmelCase__					(						self  :  List[Any]      ,				snake_case__  :  Dict ):
								"""simple docstring"""







								__lowerCAmelCase   ,						__lowerCAmelCase         =							self.fetch_inputs(features["input_ids"] )
								__lowerCAmelCase         =							{
								    "input_ids": jnp.array(snake_case__      ,				dtype=jnp.intaa ),
								    "attention_mask": jnp.array(snake_case__      ,				dtype=jnp.intaa ),
								    "start_labels": jnp.array(features["start_token"]      ,				dtype=jnp.intaa ),
								    "end_labels": jnp.array(features["end_token"]      ,				dtype=jnp.intaa ),
								    "pooled_labels": jnp.array(features["category"]      ,				dtype=jnp.intaa ),
								}
								return batch







							def       UpperCAmelCase__					(						self  :  Optional[int]      ,				snake_case__  :  list ):
								"""simple docstring"""







								__lowerCAmelCase         =							[self._fetch_inputs(snake_case__ ) for ids in input_ids]
								return zip(*snake_case__ )







							def       UpperCAmelCase__					(						self  :  Union[str, Any]      ,				snake_case__  :  list ):
								"""simple docstring"""







								__lowerCAmelCase         =							[1 for _ in range(len(snake_case__ ) )]
								while len(snake_case__ ) < self.max_length:
									input_ids.append(self.pad_id )
									attention_mask.append(0 )
								return input_ids, attention_mask






def 		_UpperCAmelCase   (	UpperCamelCase:     Tuple			,							UpperCamelCase:     Dict			,							UpperCamelCase:     Optional[Any]=None  ):






	"""simple docstring"""


	if seed is not None:
		__lowerCAmelCase         =							dataset.shuffle(seed=UpperCamelCase  )
	for i in range(len(UpperCamelCase  ) // batch_size  ):
		__lowerCAmelCase         =							dataset[i * batch_size : (i + 1) * batch_size]
		yield dict(UpperCamelCase  )






@partial(jax.pmap			,							axis_name="batch"  )
def 		_UpperCAmelCase   (	UpperCamelCase:     Optional[int]			,							UpperCamelCase:     Union[str, Any]			,							**UpperCamelCase:     List[str]  ):






	"""simple docstring"""


	def loss_fn(UpperCamelCase:     Dict  ):
		__lowerCAmelCase         =							model_inputs.pop("start_labels"  )
		__lowerCAmelCase         =							model_inputs.pop("end_labels"  )
		__lowerCAmelCase         =							model_inputs.pop("pooled_labels"  )

		__lowerCAmelCase         =							state.apply_fn(**UpperCamelCase			,							params=UpperCamelCase			,							dropout_rng=UpperCamelCase			,							train=UpperCamelCase  )
		__lowerCAmelCase   ,						__lowerCAmelCase   ,						__lowerCAmelCase         =							outputs

		return state.loss_fn(
		    UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase			,							)

	__lowerCAmelCase   ,						__lowerCAmelCase         =							jax.random.split(UpperCamelCase  )
	__lowerCAmelCase         =							jax.value_and_grad(UpperCamelCase  )
	__lowerCAmelCase   ,						__lowerCAmelCase         =							grad_fn(state.params  )
	__lowerCAmelCase         =							jax.lax.pmean({"loss": loss}			,							axis_name="batch"  )
	__lowerCAmelCase         =							jax.lax.pmean(UpperCamelCase			,							"batch"  )

	__lowerCAmelCase         =							state.apply_gradients(grads=UpperCamelCase  )
	return state, metrics, new_drp_rng






@partial(jax.pmap			,							axis_name="batch"  )
def 		_UpperCAmelCase   (	UpperCamelCase:     Optional[Any]			,							**UpperCamelCase:     List[str]  ):






	"""simple docstring"""


	__lowerCAmelCase         =							model_inputs.pop("start_labels"  )
	__lowerCAmelCase         =							model_inputs.pop("end_labels"  )
	__lowerCAmelCase         =							model_inputs.pop("pooled_labels"  )

	__lowerCAmelCase         =							state.apply_fn(**UpperCamelCase			,							params=state.params			,							train=UpperCamelCase  )
	__lowerCAmelCase   ,						__lowerCAmelCase   ,						__lowerCAmelCase         =							outputs

	__lowerCAmelCase         =							state.loss_fn(UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase  )
	__lowerCAmelCase         =							jax.lax.pmean({"loss": loss}			,							axis_name="batch"  )
	return metrics

class       a		(   train_state.TrainState					):
							lowercase_		:						Callable   							=  struct.field(pytree_node=__UpperCAmelCase					)

@dataclass
class       a		:
							lowercase_		:						Args
							lowercase_		:						Callable
							lowercase_		:						Callable
							lowercase_		:						Callable
							lowercase_		:						Callable
							lowercase_		:						wandb
							lowercase_		:						Callable   							=  None







							def       UpperCAmelCase__					(						self  :  Tuple      ,				snake_case__  :  int      ,				snake_case__  :  List[Any]      ,				snake_case__  :  Any      ,				snake_case__  :  str=None ):
								"""simple docstring"""







								__lowerCAmelCase         =							model.params
								__lowerCAmelCase         =							TrainState.create(
								    apply_fn=model.__call__      ,				params=snake_case__      ,				tx=snake_case__      ,				loss_fn=snake_case__      ,				)
								if ckpt_dir is not None:
									__lowerCAmelCase   ,						__lowerCAmelCase   ,						__lowerCAmelCase   ,						__lowerCAmelCase   ,						__lowerCAmelCase         =							restore_checkpoint(snake_case__      ,				snake_case__ )
									__lowerCAmelCase         =							{
									    "lr": args.lr,
									    "init_lr": args.init_lr,
									    "warmup_steps": args.warmup_steps,
									    "num_train_steps": num_train_steps,
									    "weight_decay": args.weight_decay,
									}
									__lowerCAmelCase   ,						__lowerCAmelCase         =							build_tx(**snake_case__ )
									__lowerCAmelCase         =							train_state.TrainState(
									    step=snake_case__      ,				apply_fn=model.__call__      ,				params=snake_case__      ,				tx=snake_case__      ,				opt_state=snake_case__      ,				)
									__lowerCAmelCase         =							args
									__lowerCAmelCase         =							data_collator
									__lowerCAmelCase         =							lr
									__lowerCAmelCase         =							params
								__lowerCAmelCase         =							jax_utils.replicate(snake_case__ )
								return state







							def       UpperCAmelCase__					(						self  :  str      ,				snake_case__  :  int      ,				snake_case__  :  Union[str, Any]      ,				snake_case__  :  Any ):
								"""simple docstring"""







								__lowerCAmelCase         =							self.args
								__lowerCAmelCase         =							len(snake_case__ ) // args.batch_size

								__lowerCAmelCase         =							jax.random.PRNGKey(0 )
								__lowerCAmelCase         =							jax.random.split(snake_case__      ,				jax.device_count() )
								for epoch in range(args.max_epochs ):
									__lowerCAmelCase         =							jnp.array(0      ,				dtype=jnp.floataa )
									__lowerCAmelCase         =							get_batched_dataset(snake_case__      ,				args.batch_size      ,				seed=snake_case__ )
									__lowerCAmelCase         =							0
									for batch in tqdm(snake_case__      ,				total=snake_case__      ,				desc=F"Running EPOCH-{epoch}" ):
										__lowerCAmelCase         =							self.data_collator(snake_case__ )
										__lowerCAmelCase   ,						__lowerCAmelCase   ,						__lowerCAmelCase         =							self.train_step_fn(snake_case__      ,				snake_case__      ,				**snake_case__ )
										running_loss += jax_utils.unreplicate(metrics["loss"] )
										i += 1
										if i % args.logging_steps == 0:
											__lowerCAmelCase         =							jax_utils.unreplicate(state.step )
											__lowerCAmelCase         =							running_loss.item() / i
											__lowerCAmelCase         =							self.scheduler_fn(state_step - 1 )

											__lowerCAmelCase         =							self.evaluate(snake_case__      ,				snake_case__ )
											__lowerCAmelCase         =							{
											    "step": state_step.item(),
											    "eval_loss": eval_loss.item(),
											    "tr_loss": tr_loss,
											    "lr": lr.item(),
											}
											tqdm.write(str(snake_case__ ) )
											self.logger.log(snake_case__      ,				commit=snake_case__ )

										if i % args.save_steps == 0:
											self.save_checkpoint(args.save_dir + F"-e{epoch}-s{i}"      ,				state=snake_case__ )







							def       UpperCAmelCase__					(						self  :  List[Any]      ,				snake_case__  :  Any      ,				snake_case__  :  Optional[Any] ):
								"""simple docstring"""







								__lowerCAmelCase         =							get_batched_dataset(snake_case__      ,				self.args.batch_size )
								__lowerCAmelCase         =							len(snake_case__ ) // self.args.batch_size
								__lowerCAmelCase         =							jnp.array(0      ,				dtype=jnp.floataa )
								__lowerCAmelCase         =							0
								for batch in tqdm(snake_case__      ,				total=snake_case__      ,				desc="Evaluating ... " ):
									__lowerCAmelCase         =							self.data_collator(snake_case__ )
									__lowerCAmelCase         =							self.val_step_fn(snake_case__      ,				**snake_case__ )
									running_loss += jax_utils.unreplicate(metrics["loss"] )
									i += 1
								return running_loss / i







							def       UpperCAmelCase__					(						self  :  Union[str, Any]      ,				snake_case__  :  Optional[int]      ,				snake_case__  :  Any ):
								"""simple docstring"""







								__lowerCAmelCase         =							jax_utils.unreplicate(snake_case__ )
								print(F"SAVING CHECKPOINT IN {save_dir}"      ,				end=" ... " )
								self.model_save_fn(snake_case__      ,				params=state.params )
								with open(os.path.join(snake_case__      ,				"opt_state.msgpack" )      ,				"wb" ) as f:
									f.write(to_bytes(state.opt_state ) )
								joblib.dump(self.args      ,				os.path.join(snake_case__      ,				"args.joblib" ) )
								joblib.dump(self.data_collator      ,				os.path.join(snake_case__      ,				"data_collator.joblib" ) )
								with open(os.path.join(snake_case__      ,				"training_state.json" )      ,				"w" ) as f:
									json.dump({"step": state.step.item()}      ,				snake_case__ )
								print("DONE" )






def 		_UpperCAmelCase   (	UpperCamelCase:     str			,							UpperCamelCase:     List[Any]  ):






	"""simple docstring"""


	print(F"RESTORING CHECKPOINT FROM {save_dir}"			,							end=" ... "  )
	with open(os.path.join(UpperCamelCase			,							"flax_model.msgpack"  )			,							"rb"  ) as f:
		__lowerCAmelCase         =							from_bytes(state.params			,							f.read()  )

	with open(os.path.join(UpperCamelCase			,							"opt_state.msgpack"  )			,							"rb"  ) as f:
		__lowerCAmelCase         =							from_bytes(state.opt_state			,							f.read()  )

	__lowerCAmelCase         =							joblib.load(os.path.join(UpperCamelCase			,							"args.joblib"  )  )
	__lowerCAmelCase         =							joblib.load(os.path.join(UpperCamelCase			,							"data_collator.joblib"  )  )

	with open(os.path.join(UpperCamelCase			,							"training_state.json"  )			,							"r"  ) as f:
		__lowerCAmelCase         =							json.load(UpperCamelCase  )
	__lowerCAmelCase         =							training_state["step"]

	print("DONE"  )
	return params, opt_state, step, args, data_collator






def 		_UpperCAmelCase   (	UpperCamelCase:     Any			,							UpperCamelCase:     Dict			,							UpperCamelCase:     Tuple			,							UpperCamelCase:     Dict  ):






	"""simple docstring"""


	__lowerCAmelCase         =							num_train_steps - warmup_steps
	__lowerCAmelCase         =							optax.linear_schedule(init_value=UpperCamelCase			,							end_value=UpperCamelCase			,							transition_steps=UpperCamelCase  )
	__lowerCAmelCase         =							optax.linear_schedule(init_value=UpperCamelCase			,							end_value=1e-7			,							transition_steps=UpperCamelCase  )
	__lowerCAmelCase         =							optax.join_schedules(schedules=[warmup_fn, decay_fn]			,							boundaries=[warmup_steps]  )
	return lr







def 		_UpperCAmelCase   (	UpperCamelCase:     Union[str, Any]			,							UpperCamelCase:     str			,							UpperCamelCase:     List[Any]			,							UpperCamelCase:     Optional[int]			,							UpperCamelCase:     Union[str, Any]  ):






	"""simple docstring"""


	def weight_decay_mask(UpperCamelCase:     int  ):
		__lowerCAmelCase         =							traverse_util.flatten_dict(UpperCamelCase  )
		__lowerCAmelCase         =							{k: (v[-1] != "bias" and v[-2:] != ("LayerNorm", "scale")) for k, v in params.items()}
		return traverse_util.unflatten_dict(UpperCamelCase  )

	__lowerCAmelCase         =							scheduler_fn(UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase			,							UpperCamelCase  )

	__lowerCAmelCase         =							optax.adamw(learning_rate=UpperCamelCase			,							weight_decay=UpperCamelCase			,							mask=UpperCamelCase  )
	return tx, lr






