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In distributed training using data parallelism, the gradient of the loss function, $$L$$, with respect to the parameters, $$\theta_t$$, for a complete mini-batch, $$\mathcal{D}_{\mathrm{mini}}$$, is computed by summing the gradients from multiple workers. Each worker calculates the gradient on a separate partition of the mini-batch, denoted as $$\mathcal{D}^1, \mathcal{D}^2, \dots, \mathcal{D}^N$$. This aggregation of gradients is represented by the formula:

$$\frac{\partial L_{\theta_t}(\mathcal{D}_{\mathrm{mini}})}{\partial \theta_t} = \underbrace{\frac{\partial L_{\theta_t}(\mathcal{D}^1)}{\partial \theta_t}}_{\textrm{worker 1}} + \underbrace{\frac{\partial L_{\theta_t}(\mathcal{D}^2)}{\partial \theta_t}}_{\textrm{worker 2}} + \cdots + \underbrace{\frac{\partial L_{\theta_t}(\mathcal{D}^N)}{\partial \theta_t}}_{\textrm{worker } N}$$

This allows for parallel computation, significantly speeding up the training process for large models.

Distributed Gradient Calculation

In a distributed training setup, a mini-batch of data is split between two workers. Each worker computes a gradient vector for a model's two parameters based on its portion of the data. Given the results below, what is the final aggregated gradient vector that will be used to update the model's parameters? Explain the simple mathematical operation used to combine the worker gradients.

Aggregated Gradient Calculation

A machine learning model is being trained in a distributed environment using two workers. For a single parameter, worker 1 computes a gradient of `+0.5` on its partition of the data, and worker 2 computes a gradient of `-0.2` on its partition. What is the value of the final, aggregated gradient that will be used to update this parameter for the entire mini-batch?

In a data-parallel distributed training setup with four workers, a mini-batch is split equally among them. For a particular training step, the gradient vectors calculated on three of the workers have a similar, small magnitude. However, the fourth worker calculates a gradient vector with a magnitude ten times larger than the others, possibly due to a corrupted data sample. According to the standard aggregation method for this setup, what is the most likely effect on the combined gradient used to

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