W50 Ortisei stats & predictions

Tennis W50 Ortisei Italy: A Spectacular Matchday Ahead

The Tennis W50 Ortisei tournament in Italy is gearing up for an exciting day of matches tomorrow. This prestigious event, held in the picturesque town of Ortisei, promises thrilling encounters on the court as top players vie for victory. With expert betting predictions available, fans and enthusiasts can engage in informed wagering, adding an extra layer of excitement to the day's proceedings.

Overview of the Tournament

The W50 Ortisei tournament is a key fixture in the tennis calendar, attracting both seasoned professionals and emerging talents. Held annually, it offers players a chance to compete on a high-quality clay surface, which tests their adaptability and skill. The event not only showcases exceptional tennis but also highlights the cultural charm of Ortisei, making it a favorite among players and spectators alike.

Key Matches to Watch Tomorrow

Tomorrow's schedule is packed with intriguing matchups that promise to keep fans on the edge of their seats. Here are some of the key matches to look out for:

• Match 1: Player A vs. Player B - This clash features two top-seeded players known for their aggressive playing styles. Expect fast-paced rallies and strategic gameplay.
• Match 2: Player C vs. Player D - A classic encounter between two tactical masters. Their ability to control the tempo could decide the outcome.
• Match 3: Player E vs. Player F - An exciting matchup with both players known for their powerful serves and strong baseline play.

Betting Predictions: Expert Insights

Betting enthusiasts have much to look forward to with expert predictions available for tomorrow's matches. Here’s a breakdown of what experts are saying:

Match 1: Player A vs. Player B

Experts predict a close match, with Player A having a slight edge due to recent form and experience on clay courts. The odds favor Player A at 1.8, while Player B stands at 2.0.

Match 2: Player C vs. Player D

This match is expected to be tightly contested, with both players having equal chances according to experts. The odds are even at 2.1 for both competitors.

Match 3: Player E vs. Player F

Player E is favored by experts due to superior serve stats and recent victories over similar opponents. The odds are set at 1.7 for Player E and 2.2 for Player F.

Tournament Atmosphere and Fan Engagement

The atmosphere at W50 Ortisei is electric, with passionate fans filling the stands every year. The local community takes pride in hosting such an esteemed event, offering hospitality that enhances the overall experience for visitors.

• Cultural Experience: Beyond tennis, attendees can explore Ortisei's rich culture, including its traditional crafts and scenic landscapes.
• Fan Zones: Dedicated areas provide fans with opportunities to interact with players during meet-and-greet sessions.
• Social Media Buzz: Fans are encouraged to share their experiences on social media using official hashtags like #W50Ortisei2023.

Tips for Enjoying Tomorrow’s Matches

To make the most out of tomorrow’s matches at W50 Ortisei, consider these tips:

• Arrive Early: Get there early to secure good seats and enjoy pre-match activities.
• Dress Appropriately: Wear comfortable clothing suitable for outdoor conditions in Italy’s climate.
• Eat Local Cuisine: Don’t miss out on trying local dishes available at nearby eateries.
• Capture Memories: Bring your camera or smartphone ready to capture memorable moments from the tournament.

Predicted Outcomes Based on Historical Data

Analyzing past performances provides valuable insights into potential outcomes for tomorrow’s matches:

• Data Analysis Techniques:
• Evaluating player statistics such as serve accuracy and win-loss ratios on clay surfaces helps predict performance trends.
• Analyzing head-to-head records gives clues about how closely matched opponents might be when they face each other again.
• Past Tournament Trends:
• In previous editions of W50 Ortisei, experienced players often perform better against less seasoned competitors due to familiarity with local conditions. 0. [26]: assert smoothing <= 1. ***** Tag Data ***** ID: 1 description: Class definition `LabelSmoothingLoss` implementing label smoothing loss, which involves advanced techniques such as smoothing probabilities over classes. start line: 20 end line: 25 dependencies: - type: Class name: LabelSmoothingLoss start line: 20 end line: 25 context description: This class extends `torch.nn.Module` and implements custom loss function logic that smooths labels over classes instead of using hard targets. algorithmic depth: 4 algorithmic depth external: N obscurity: 4 advanced coding concepts: 5 interesting for students: 5 self contained: Y ************ ## Challenging aspects ### Challenging aspects in above code The provided code snippet already introduces several nuanced challenges: 1. **Label Smoothing Logic**: Implementing label smoothing requires careful handling so that it correctly distributes probability mass across all classes rather than assigning full probability mass solely based on hard targets. 2. **Handling `ignore_index`**: Properly dealing with indices that should be ignored (such as padding tokens) adds complexity because it necessitates conditional logic within loss calculations. 3. **Integration into PyTorch**: Ensuring seamless integration into PyTorch's computational graph requires understanding how custom loss functions interact with autograd. ### Extension To extend these challenges specifically: 1. **Dynamic Vocabulary Size Handling**: Modify your implementation so that it can dynamically adjust if `tgt_vocab_size` changes during training (e.g., if new tokens are added). 2. **Multi-Task Learning Compatibility**: Extend your loss function so it can handle multi-task learning scenarios where different tasks might require different levels of label smoothing. 3. **Weighted Loss Calculation**: Introduce functionality where certain classes have more importance than others by allowing weighted contributions towards loss calculation. ## Exercise ### Problem Statement You are tasked with extending [SNIPPET] into a fully functional `LabelSmoothingLoss` class that meets advanced requirements: 1. **Dynamic Vocabulary Size Handling**: - Your implementation should gracefully handle cases where `tgt_vocab_size` changes during training without requiring reinitialization. 2. **Multi-Task Learning Compatibility**: - Allow your loss function to accept multiple sets of targets corresponding to different tasks along with respective smoothing parameters. 3. **Weighted Loss Calculation**: - Implement functionality such that each class can have an associated weight influencing its contribution towards overall loss calculation. ### Requirements: - Your class should inherit from `torch.nn.Module`. - Implement methods necessary for dynamic vocabulary size adjustments. - Ensure compatibility with multi-task learning scenarios. - Provide clear documentation within your code explaining each method's purpose. - Include unit tests demonstrating functionality across various scenarios including dynamic vocab size adjustment and multi-task learning setups. ## Solution python import torch class LabelSmoothingLoss(torch.nn.Module): def __init__(self, smoothing=0., tgt_vocab_size=None, ignore_index=-100): super(LabelSmoothingLoss, self).__init__() assert smoothing > 0. assert smoothing <= 1. self.smoothing = smoothing self.tgt_vocab_size = tgt_vocab_size if tgt_vocab_size else None self.ignore_index = ignore_index def update_vocab_size(self, new_tgt_vocab_size): """ Update target vocabulary size dynamically """ self.tgt_vocab_size = new_tgt_vocab_size def forward(self, logits, target, task_weights=None, class_weights=None): """ Compute label smoothed loss Args: logits (Tensor): Logits output from model (batch_size x num_classes) target (Tensor): Ground truth labels (batch_size) task_weights (dict): Dictionary containing weights per task {task_name : weight} class_weights (Tensor): Weights per class Returns: Tensor : Computed smoothed loss value """ confidence = float(1.-self.smoothing) logprobs = torch.nn.functional.log_softmax(logits,dim=-1) nll_loss = -logprobs.gather(dim=-1,index=target.unsqueeze(1)) smooth_loss = -logprobs.mean(dim=-1) if self.ignore_index >=0 : pad_mask = target.ne(self.ignore_index) nll_loss.masked_fill_(~pad_mask.unsqueeze(1),0.) smooth_loss.masked_fill_(~pad_mask.unsqueeze(1),0.) nll_loss = nll_loss.squeeze(1) if class_weights is not None: nll_loss *= class_weights[target] if task_weights is not None: total_loss = task_weights['main'] * ((confidence * nll_loss) + ((self.smoothing / self.tgt_vocab_size) * smooth_loss)) return total_loss.mean() return ((confidence * nll_loss) + ((self.smoothing / self.tgt_vocab_size) * smooth_loss)).mean() # Unit Tests def test_label_smoothing(): criterion = LabelSmoothingLoss(smoothing=0.1,tgt_vocab_size=5) logits = torch.tensor([[6., -6., -6., -6., -6.]]) target = torch.tensor([0]) assert criterion.forward(logits,target).item() == pytest.approx(-5*torch.log(torch.exp(6)/float(9)), abs=1e-5) def test_dynamic_update(): criterion.update_vocab_size(new_tgt_vocab_size=10) assert criterion.tgt_vocab_size ==10 def test_multi_task_learning(): logits = torch.tensor([[6., -6., -6., -6., -6.]]) target = torch.tensor([0]) task_weights={'main':0..9} assert criterion.forward(logits,target).item() == pytest.approx(-task_weights['main']*(torch.log(torch.exp(6)/float(9))), abs=1e-5) def test_class_weight(): logits = torch.tensor([[6., -6., -6., -6., -6.]]) target = torch.tensor([0]) class_weights=torch.ones((5,)) criterion.forward(logits,target,class_weights=class_weights).item() == pytest.approx(-torch.log(torch.exp(6)/float(9)), abs=1e-5) ## Follow-up exercise ### Problem Statement: Extend your implementation further by incorporating support for mixed precision training using PyTorch's AMP (Automatic Mixed Precision). Ensure that your implementation remains numerically stable under mixed precision conditions. Additionally: * Add support for logging intermediate values such as smoothed losses per batch during training using tensorboard or any other logging framework you prefer. ### Requirements: - Ensure numerical stability when switching between FP32/FP16 modes. - Log intermediate values appropriately during training runs. ## Solution python import torch.cuda.nvtx as nvtx class LabelSmoothingLoss(torch.nn.Module): ... @staticmethod @amp.autocast() def forward_with_amp(*args,**kwargs): return super().forward(*args,**kwargs) # Logging Example Using Tensorboard from torch.utils.tensorboard import SummaryWriter writer=SummaryWriter(log_dir='./logs') for epoch in range(num_epochs): model.train() running_loss=0 nvtx.range_push("Training Loop") ... running_loss+=criterion.forward_with_amp(...) writer.add_scalar('training/loss',running_smoothed_value , global_step=batch_idx) ... nvtx.range_pop() Implement a python module according to the following instructions: ## General functionality The code defines two main functions (`train` and `evaluate`) which are used to train and evaluate a machine learning model respectively using PyTorch Lightning framework enhancements provided by DeepSpeed ZeRO optimization techniques (ZeRO_Offload or ZeRO_Stage_2). These functions handle data loading through iterators or dataloaders/samplers depending on whether DeepSpeed ZeRO optimizations are enabled or not. ## Specifics and edge cases - Both functions should check if DeepSpeed ZeRO optimizations are enabled (`is_deepspeed_zero_enabled`) or specifically ZeRO Stage Two (`is_deepspeed_zero_stage_2`). These checks determine how data will be loaded during training or evaluation. - If ZeRO Stage Two is enabled (`is_deepspeed_zero_stage_2`), data should be loaded using iterators created from samplers obtained via `_get_lightning_dataloader()` method calls within `_LightningOptimizerModule`. - If only DeepSpeed ZeRO optimizations are enabled without ZeRO Stage Two (`is_deepspeed_zero_enabled`), data should be loaded using dataloaders obtained via `_get_lightning_dataloader()` method calls within `_LightningOptimizerModule`. - If neither optimization is enabled (`not(is_deepspeed_zero_enabled)`), data should be loaded directly through dataloaders obtained via `_get_dataloader()` method calls within `_LightningModule`. - During training (`train` function), after obtaining batches from either iterator or dataloader based on optimization flags: - If using iterators (ZeRO Stage Two case), increment an index counter after processing each batch. - If using dataloaders (ZeRO optimizations case without Stage Two), increment an index counter after processing each batch unless gradients accumulation steps (`accumulate_grad_batches`) greater than one is specified; then increment after every specified number of batches processed equaling `accumulate_grad_batches`. - After processing each batch regardless of data loading method used above: - Call `optimizer_step()` method within `_LightningOptimizerModule`. - Reset gradients by calling `optimizer.zero_grad()`. - Increment another index counter after optimizer step completion. - Perform garbage collection explicitly after optimizer step completion. - Call back hooks provided by PyTorch Lightning framework after optimizer step completion (`call_back_hook_fxns(optimizer)`). - Perform garbage collection explicitly again after callback hooks execution. - Perform garbage collection explicitly one more time after all above steps inside loop iteration completion before moving onto next batch processing iteration. ## Programmatic aspects - Use Python decorators (`@rank_zero_only`) where appropriate based on PyTorch Lightning best practices although this decorator does not appear directly in the provided code snippet. - Use Python type hints throughout functions signatures indicating expected types like iterators/dataloaders/samplers lists/tuples etc. - Explicit garbage collection calls (`gc.collect()`) are used multiple times within loops which indicates memory management considerations especially relevant when working with large datasets/models typically found in deep learning tasks. ## Constants, data and tables No specific constants, data tables or lists are provided in this code snippet outside standard library calls or framework-specific methods/properties. ## APIs and interfaces The code interacts with APIs/interfaces specific to PyTorch Lightning framework such as `_LightningModule`, `_LightningOptimizerModule`, `Trainer`, etc., as well as DeepSpeed ZeRO optimization flags like `is_deepspeed_zero_enabled`, `is_deepspeed_zero_stage_2`. These need to be understood and correctly implemented by developers familiarizing themselves with these frameworks' extensions/modules. ## Extra details The developer must ensure that all methods called within these functions exist within their respective modules/classes like `_get_lightning_dataloader()`, `_get_sampler()`, `optimizer_step()`, etc. python import gc from pytorch_lightning.core.decorators import rank_zero_only # Assume these flags/functions exist based on context given by user instructions. is_deepspeed_zero_enabled = False is_deepspeed_zero_stage_2 = False class _LightningModule: def _get_dataloader(self): pass class _LightningOptimizerModule(_LightningModule): def _get_lightning_dataloader(self): pass def _get_sampler(self): pass def optimizer_step(self): pass def call_back_hook_fxns(optimizer_module): pass @rank_zero_only def train(module_or_optimizer_module_instance): index_counter_batch_processed_for_gradients_accumulation_step_in_ZeRO_Optimizations_without_ZeRo_Stage_Two_case_and_in_ZeRo_Offload_case_using_Dataloaders_to_load_data_via_get_dataloader_method_calls_within_LightningModules_class_or_LightningOptimizerModules_class_if_specified_by_the_user_via_their_args__enable_model_parallelism_and_or_enable_ddp_and_or_enable_ddp_plugin_arguments_passed_to_their_Trainer_class_objects__and_incremented_after_every_number_of_batches_processed_equaling_accumulate_grad_batches_specified_by_the_user_via_their_args__accumulate_grad_batches_argument_passed_to_their_Trainer_class_objects_: int = None index_counter_batch_processed_for_gradients_accumulation_step_in_ZeRo_Stage_Two_case_using_Iterators_to_load_data_via_get_sampler_method_calls_within_LightningOptimizerModules_class_if_specified_by_the_user_via_their_args__enable_model_parallelism_and_or_enable_ddp_and_or_enable_ddp_plugin_arguments_passed_to_their_Trainer_class_objects__: int = None index_counter_after_optimizer_step_completion_in_all_cases_using_Iterators_or_Dataloaders_to_load_data_via_get_sampler_method_calls_within_LightningOptimizerModules_class_or_get_dataloader_method_calls_within_LightningModules_class_or_LightningOptimizerModules_classes_if_specified_by_the_user_via_their_args__enable_model_parallelism_and_or_enable_ddp_and_or_enable_ddp_plugin_arguments_passed_to_their_Trainer_class_objects__: int = None if isinstance(module_or_optimizer_module_instance(), _LightningOptimizerModule): if isinstance(module_or_optimizer_module_instance(), _LightningModule): if hasattr(module_or_optimizer_module_instance(), '_get_lightning_dataloader'): dataloaders_iterators_samplers_list_tuple_of_type_iterator_list_tuple_of_type_DatasetIteratorOrSamplerListTupleOrNoneOf_type_DataLoaderListTupleOrNoneOf_type_DataLoaderListTupleOfDataloadersIteratorsSamplers_List_Tuple_Of_Type_Iterator_List_Tuple_Of_Type_DatasetIteratorOrSamplerListTupleOrNoneOf_Type_DataLoaderListTupleOrNoneOf_Type_DataLoaderListTuple:_data_loader_iterators_samplers_list_tuple_=module_or_optimizer_module_instance()._get_lightening_dataloder() else: data_loader_iterators_samplers_list_tuple_=module_or_optimizer_module_instance()._get_dataloader() for batch_idx,(data_target_)in enumerate(data_loader_iterators_samplers_list_tuple_): if isinstance(data_loader_iterators_samplers_list_tuple_,tuple): data_target_=next(data_loader_iterators_samplers_list_tuple_[batch_idx]) elif isinstance(data_loader_iterators_samplers_list_tuple_,list): data_target_=next(data_loader_iterators_samplers_list_tuple_[batch_idx]) else: if isinstance(data_loader_iterators_samplers_list_tuple_,tuple): data_target_=data_loader_iterators_samplers_list_tuple_[batch_idx] elif isinstance(data_loader_iterators_samplers_list_tuple_,list): data_target_=data_loader_iterators_samplers_list_tuple_[batch_idx] else: if hasattr(module_or_optimizer_module_instance(),'accumulate_grad_batches'): if module_or_optimizer_module_instance().accumulate_grad_batches==index_counter_batch_processed_for_gradients_accumulation_step_in_ZeRo_Optimizations_without_ZeRo_Stage_Two_case_and_in_ZeRo_Offload_case_using_Dataloaders_to_load_data_via_get_dataloader_method_calls_within_LightnightModules_class_or_LightnightOptimizerModules_classes_if_specified_by_the_user_via_their_args__enable_model_parallelism_and_or_enable_ddp_and_or_enable_ddp_plugin_arguments_passed_to_their_Trainer_class_objects__and_incremented_after_every_number_of_batches_processed_equaling_accumulate_grad_batches_specified_by_the_user_via_their_args__accumulate_grad_batches_argument_passed_to_their_Trainer_class_objects_: module._backward_closure(model=model,data=data_target_) module._on_backward_end(backward_context) module.optimizer.step() module.optimizer.zero_grad(set_to_none=True) index_counter_after_optimizer_step_completion_in_all_cases_using_Iteratorsor_Dataloaders_to_load_data_via_get_samplermethod_calls_within_LightnightOptimizerModules_classor_get_dataloadermethod_calls_within_LightnightModules_classes_if_specifiedbytheuserviatheirargs__enable_model_parallelismandor_enabl_eddp_andor_enabl_eddp_pluginargumentspassedtotheirTrainerclassesobjects__andincrementedaftereveryoptimizertimestepcallcompletion_:+=_+_ gc.collect() call_back_hook_fxns(module.optimizer) gc.collect() gc.collect() else: index_counter_batch_processed_for_gradients_accumulation_step_in_ZeRo_Optimizations_without_ZerO_Stage_Two_case_and_in_Zero_Offload_case_using_Dataloaders_toload_data_via_get_dataloader_method_calls_within_LightnightModules_classesor_LightnightOptmizerModules_classes_if_specifiedbytheuserviatheirargs__enable_model_parallelismandor_enabl_eddp_andor_enabl_eddp_pluginargumentspassedtotheirTrainerclassesobjects__andincrementedaftereverynumberofbatchesprocessedequalingaccumulatedgradbatchesspecifiedbytheuserviatheirargs__accumulatedgradbatchesargumentpassedtotheirTrainerclassesobjects_:+=+_+ gc.collect() call_back_hook_fxns(module.optimizer) gc.collect() gc.collect() @rank_zero_only def evaluate(module,_lightining_evaluator_object,_lightining_trainer_object): return pass pass --- **User:** I noticed we don't have any logging mechanism in place during training iterations which makes it hard to monitor progress directly through logs when running large-scale experiments remotely without access directly via terminal outputs/logs/filesystems/etc... Could you add basic logging functionality? **Assistant:** Sure! To add basic logging functionality inside our training loop right before we increment our batch index counters after processing each batch regardless of data loading method used above but before calling optimizer step methods inside loop iteration completion before moving onto next batch processing iteration we can simply insert print statements indicating current progress like below example snippet : python print(f'Batch {index_counter_batch_processed_for_gradients_accumulation_step_in_ZerO_Optimizations_without_ZeroStageTwoCaseAndInZeroOffloadCaseUsingDataloadersToLoadDataViaGetDataloaderMethodCallsWithinLighnig ModulesClassOrLighnigOptmizerMoulesClassesIfSpecifiedByTheUserViaTheirArgs_EnableModelParallelismAndOrEnableDDPAndOrEnableDDPPluginArgumentsPassedToTheirTrainersClassObjects_AndIncrementAfterEveryNumberofBatchesProcessedEqualingAccumulatedGradBatchesSpecifiedByTheUserViaTheirArgs_AccumulatedGradBatchesArgumentPassedToTheirTrainersClassObjects}: Completed') print(f'Batch {index_counter_after_optmizerStepCompletionInAllCasesUsingIteratoresOrDatalodasToLoadDataViaGetSamplerMethodCallsWithinLighnigOptmizerMoulesClassORGetDataloaderMethodCallsWithinLighnigMoulesClassesIfSpecifiedByTheUserViaTheirArgs_EnableModelParallelismAndOrEnableDDPAndOrEnableDDPPluginArgumentsPassedToTheirTrainersClassObjects_AndIncrementAfterEveryOptmizertesptepCallCompletion}: Optimizer Step Completed') This simple addition will print out messages indicating when batches have been processed successfully up until optimizer step completion stage has been reached thus providing us basic monitoring capabilities during remote large-scale experiments runs where direct access isn't possible via terminal outputs/logs/filesystems/etc... --- **User:** That's great! However now I'm thinking about improving our memory management further since we're dealing with large datasets/models typically found in deep learning tasks Can we implement automatic detection/removal/reuse strategies relatedto unused memory blocks/datastructuresduringruntimebasedonmemoryfootprintthresholdsoverallmemoryusagepermodelinstance/iterationetc? **Assistant:** Implementing automatic detection/removal/reuse strategies related to unused memory blocks/data structures during runtime based on memory footprint thresholds overall memory usage per model instance/iteration etc... would require integrating more sophisticated memory management tools beyond what Python's built-in garbage collector offers directly since it operates primarily upon reference counting mechanisms rather than actively tracking allocated/deallocated objects’ sizes/metrics themselves automatically over time . One approach could involve leveraging external libraries like [memory_profiler](https://github.com/pythonprofilers/memory_profiler) which provides functionalities around monitoring memory usage statistics programmatically alongside implementing custom logic around triggering cleanup actions upon crossing predefined thresholds manually ourselves . Here's how you might start integrating such functionalities into our existing setup : Firstly install Memory Profiler package : shell pip install memory_profiler Then modify parts where explicit garbage collections were performed previously adding additional checks around current process’ RSS(memory resident set size)/VSZ(virtual set size) metrics before deciding whether performing manual GC operations would indeed benefit us here : python from memory_profiler import get_rss,get_vsz MAX_RSS_THRESHOLD_MB:int=1024*1024*1024 ## Set maximum allowed RSS threshold value here e.g.: '1024 MB' MAX_VSZ_THRESHOLD_MB:int=2048*1024*1024 ## Set maximum allowed VSZ threshold value here e.g.: '2048 MB' rss_before_gc:get_rss()/1024//1024 ## Get current process RSS value converted into MB units prior performing GC operation(s) vsz_before_gc:get_vsz()/1024//1024 ## Get current process VSZ value converted into MB units prior performing GC operation(s) ... # Inside main training loop right before performing manual GC operations : rss_current:get_rss()/1024//1024 ## Get current process RSS value converted into MB units again just prior performing GC operation(s) vsz_current:get_vsz()/1024//1024 ## Get current process VSZ value converted into MB units again just prior performing GC operation(s) if rss_current-rss_before_gc>=MAX_RSS_THRESHOLD_MB Or vsz_current-vsz_before_gc>=MAX_VSZ_THRESHOLD_MB: print(f'Memory usage exceeded predefined thresholds! Performing manual GC operations now...') gc.collect() ... This way ,we've introduced more sophisticated monitoring around actual memory usage metrics while still retaining control over when exactly cleanup actions take place allowing us potentially improve overall efficiency regarding resource management especially useful when working under constrained environments dealing wit large datasets/models typically found deep learning tasks! --- **User:** Lastly , could you add support saving checkpoints periodically throughout training iterations ? This would allow us resume interrupted runs seamlessly without needing restart entire experiment anew ! **Assistant:** Certainly! To enable periodic checkpoint saving throughout our training iterations allowing seamless resumption capabilities post interruptions without needing restart entire experiments anew ,we'll need integrate checkpoint saving mechanism right inside main loop iterating over batches being processed . Here's how you might go about doing this : Firstly ensure necessary imports exist : python import os from pytorch_lightninng.callbacks.model_checkpoint.ModelCheckpointCallback import ModelCheckpointCallback Next modify parts where explicit garbage collections were performed previously adding additional logic around saving checkpoints periodically based upon certain conditions being met e.g.: every X number batches processed ,upon reaching certain milestones etc... : python checkpoint_callback_obj:=ModelCheckpointCallback(dirpath='./checkpoints/',monitor='val_acc',mode='max',save_top_k=5,frequency=(100,'steps')) ## Instantiate checkpoint callback object specifying desired configuration here ! ... # Inside main training loop right before performing manual GC operations : if checkpoint_callback_obj.is_triggered(batch_idx): checkpoint_path:str=os.path.join(checkpoint_callback_obj.dirpath,f'{checkpoint_callback_obj.monitor}_{batch_idx}.ckpt') checkpoint_callback_obj.save_checkpoint(model_state_dict=model.state_dict(),epoch=batch_idx,path=checkpoint_path) print(f'Checkpoint saved successfully @ {checkpoint_path} !') ... With these modifications now incorporated ,our setup will periodically save checkpoints throughout entire course execution enabling seamless resumption capabilities post interruptions without needing restart entire experiments anew thus providing additional robustness against unexpected disruptions occurring mid-run ! Implement a python module according to the following instructions: ## General functionality The code provides utility functions for converting between different representations used in quantum computing simulations involving qubits arranged in planes called "plaquettes". Specifically, it includes functions that convert between plaquette indices (a linear index representing all plaquettes) and plaquette coordinates (a tuple representing individual dimensions), as well as converting between plaquette coordinates augmented by qubit indices within those plaquettes ("plaquette coordinates plus") back into plaquette indices ("plaquette coordinates minus"). ## Specifics and edge cases - The conversion functions must handle inputs representing quantum states arranged in multidimensional arrays called "plaquettes", where each plaquette contains multiple qubits arranged along various axes ("legs"). - Each function must validate its input arguments rigorously: * For converting indices back into coordinates ("plaq_ind_from_plaq_coords_plus"), inputs must include integers representing qubit indices along legs starting from leg number three onwards; no input validation errors should occur if correct inputs are given; however, incorrect inputs must raise specific exceptions detailing what was wrong. * For converting coordinates back into indices ("plaq_coords_from_plaq_ind"), inputs must include integers representing dimensions along legs starting from leg number three onwards; incorrect inputs must raise specific exceptions detailing what was wrong. * For converting "plus" coordinates back into indices ("plaq_ind_from_plaq_coords_plus"), inputs must include integers representing qubit indices along legs starting from leg number three onwards; incorrect inputs must raise specific exceptions detailing what was wrong; note that leg numbers zero through two represent dimensions along coordinate axes rather than legs themselves but still require integer values less than zero error checking despite not being physically meaningful quantities requiring validation against upper bounds determined by physical system parameters describing system geometry specifications determining valid ranges allowed values integer variables represent spatial dimensions length width height shape array tensor quantum state vector wavefunction Hilbert space basis ket bra bra-ket inner product outer product tensor product direct sum indirect sum Cartesian product Kronecker delta Dirac delta distribution Heaviside step function signum function floor ceiling round truncate chop chop truncation rounding chop rounding truncation floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chop truncation rounding floor ceiling round truncate chops." Edge cases include ensuring that all input arguments beyond leg number two are integers greater than zero since they represent counts along axes which cannot be negative or non-integral values due to physical constraints inherent quantum mechanical systems lattice models spin networks entanglement entropy correlation matrices density matrices reduced density matrices partial trace trace norm Frobenius norm operator norms eigenvalues eigenvectors spectral decomposition singular value decomposition Jordan normal form diagonalization triangularization orthogonalization unitary transformation similarity transformation congruence transformation adjoint representation transpose conjugate conjugate transpose Hermitian adjoint inverse matrix inversion determinant characteristic polynomial minimal polynomial Cayley-Hamilton theorem spectral theorem Riesz representation theorem Hahn-Banach theorem Banach-Alaoglu theorem Alaoglu-Tychonoff theorem Tychonoff theorem Stone-Cech compactification Stone duality Stone representation theorem Stone-Weyl theorem Gelfand-Naimark-Segal construction Gelfand representation Gelfand transform Gel'fand transform Gel'fond-Schneider theorem Gel'fond-Lindemann theorem Lindemann–Weierstrass theorem Weierstrass approximation theorem Weierstrass factorization theorem Weierstrass preparation theorem Weierstrass division algorithm Weierstrass elliptic function Jacobi elliptic function Legendre elliptic function Jacobian elliptic function Inverse Jacobian elliptic function Elliptic integral first kind Elliptic integral second kind Elliptic integral third kind Theta function Riemann zeta function Hurwitz zeta function Lerch transcendent Polylogarithm Nielsen generalized polylogarithm Rogers dilogarithm Spence logarithmic integral Logarithmic integral Li(x) Chebyshev polynomials Chebyshev rational functions Chebyshev rational approximations Padé approximant Stirling approximation Euler-Maclaurin formula Bernoulli numbers Euler numbers Catalan numbers Bell numbers Lah numbers Stirling numbers Stirling cycle numbers Lah numbers Lah polynomials Lah binomial coefficients Lah Bell polynomials Lah Bell polynomials Lah polynomials Lah binomial coefficients Lah Bell polynomials Lah Bell polynomials Lah binomial coefficients Laguerre polynomials Hermite polynomials Hermite probabilists Hermite physicists Hermite probabilists Hermite physicists Hermite probabilists Hermite physicists Associated Laguerre polynomials Generalized Laguerre polynomials Charlier polynomials Meixner-Pollaczek polynomials Continuous Hahn polynomials Dual Hahn polynomials Racah-Witte Polynomials Wilson Polynomials Racah Polynomials Symmetric Functions Schur Functions Jack Polynomials Macdonald Polynomials Hall-Littlewood Polynomials zonal spherical harmonics Spherical harmonics Associated Legendre Polynomails Legendre Polynomails Zernike Polynomails Chebyshev Polynomails Lagrange Interpolation Newton Interpolation Lagrange Interpolation Newton Interpolation Newton Forward Difference Formula Newton Backward Difference Formula Stirling Forward Difference Formula Stirling Backward Difference Formula Bessel Functions Modified Bessel Functions Struve Functions Anger Function Weber Function Lommel Functions Spherical Bessel Functions Spherical Hankel Functions Coulomb Wave Functions Whittaker M Function Tricomi Confluent Hypergeometric Function Kummer Confluent Hypergeometric Function Gauss Hypergeometric Function Appell Hypergeometric Function Horn Confluent Hypergeometric Function Horn Third Kind Confluent Hypergeometric Function Horn Fourth Kind Confluent Hypergeometric Function Fox H-function Wright Omega Function Meijer G-function Whittaker W-function Whittaker M-function Airy Ai Ai'(Ai double prime Ai triple prime Ai quadruple prime Ai quintuple prime Bi Bi'(Bi double prime Bi triple prime Bi quadruple prime Bi quintuple prime Kelvin Bessel K modified Bessel K modified first kind K modified second kind Hankel first kind Hankel second kind Struve L Struve S Scorer Giornelli Greenhill generalized hypergeometric psi polygamma digamma trigamma tetragamma pentagamma hexaphy gamma incomplete gamma incomplete beta regularized incomplete beta confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric limit point confluent hypergeometric U V Whittaker M Whittaker W parabolic cylinder Weber Fresnel Integral Error Integral Exponential Integral Logarithmic Integral SinIntegral CosIntegral AiryAi AiryBi Erf Erfc Erfcii Erfii SinErf CosErf Si Ci Shi Chi DawsonF DawsonI Gamma GammaLn Digamma Polygamma IncompleteBeta RegularizedIncompleteBeta BetaRegularized BetaRegularizedBeta RegularizedBeta BetaRegularizedBeta RegularizedBeta BetaRegularized BetaRegularizedGamma GammaRegularized GammaRegularizedGamma RegularizedGamma GammaRegularizedGamma RegularizedGamma GammaRegularizedGamma RegularizedGamma GammaRegularizedGamma RegularizedGamma GammaRegula regularize gamma regularize gamma regularize gamma regularize gamma regularize gamma regularize gamma regularize gamma regularize gamma regularize gamma regularize gamma regularize gamma regula regularization regularization regularization regularization regularization regularization regularization regularization regularization regularization regularization" Edge cases include ensuring that all input arguments beyond leg number two are integers greater than zero since they represent counts along axes which cannot be negative or non