What is differentially private SGD?

As machine learning models become increasingly integrated into our lives, the need to protect users' privacy becomes more critical. These models often require vast amounts of data to make accurate predictions, which may include sensitive user information. In this Answer, we will discuss why privacy is important in training machine learning models, how to measure privacy, and introduce differentially private stochastic gradient descent (DP-SGD), which is a privacy-preserving optimization algorithm.

Role of privacy in machine learning

Machine learning models, especially deep learning models, have been proven to perform exceptionally well in various domains such as image recognition, natural language processing, and recommendation systems. However, these models can also inadvertently learn sensitive information from the training data, exposing users' private details. This exposure raises ethical concerns and legal implications, as it may lead to unintended discrimination, identity theft, and other privacy breaches.

To counter these concerns, the field of privacy-preserving machine learning has emerged, aiming to develop techniques that enable models to learn from data without revealing sensitive information about individual users.

Measuring privacy: Differential privacy

One of the most widely-accepted frameworks for measuring privacy in machine learning is differential privacy. Differential privacy provides a formal definition of privacy and quantifies the amount of information that can be revealed about an individual when they participate in a data analysis process. Mathematically, a randomized mechanism $M$ satisfies $D$ and $D'$ differing in a single element, and for all subsets of possible outputs $S \subseteq Range(M)$ , the following inequality holds:

Where $\epsilon$ and $\delta$ are non-negative privacy parameters. Smaller values of $\epsilon$ and $\delta$ indicate stronger privacy guarantees. Intuitively, this definition ensures that the presence or absence of any individual's data in the dataset has a negligible impact on the outcome of the analysis.

Differentially private stochastic gradient descent

Stochastic gradient descent (SGD) is a popular optimization algorithm used to train machine learning models. To incorporate differential privacy into SGD, the differentially private SGD (DP-SGD) was proposed. DP-SGD works by adding carefully calibrated noise to the gradients during the training process, ensuring that the final model satisfies the desired privacy guarantees.

DP-SGD can be summarized in the following steps:

1. Sample a random mini-batch $B$ from the dataset $D$.

2. Compute the gradient $\nabla L(\theta)$ for the mini-batch, where $L(\theta)$ is the loss function and $\theta$ are the model parameters.

3. Clip the gradients to a maximum norm $C$: $\nabla L_{\text{clipped}}(\theta) = \min\left(\frac{C}{|\nabla L(\theta)|}, 1\right)\nabla L(\theta)$.

4. Add noise to the clipped gradients: $\nabla L_{\text{noisy}}(\theta) = \nabla L_{\text{clipped}}(\theta) + \text{noise}$.

5. Update the model parameters: $\theta \leftarrow \theta - \eta \nabla L_{\text{noisy}}(\theta)$, where $\eta$ is the learning rate.

The noise added in step 4 is typically Gaussian or Laplacian noise, and its scale is determined by the privacy parameters $\epsilon$ and $\delta$ , as well as the dataset's sensitivity. Sensitivity is the maximum change in the output caused by the presence or absence of a single data point. By adding this noise, the influence of any single data point on the model parameters is obscured, ensuring privacy.

The choice of the clipping parameter $C$ and the noise scale is crucial in DP-SGD. If the clipping parameter is too small, the model might underfit the data, while if it is too large, the privacy guaranteed might be weakened. Similarly, privacy could be compromised if the noise scale is too small, while if it is too large, the model's utility could degrade. Therefore, it is essential to find a balance between these parameters to achieve both accurate and private models.

The privacy-accuracy trade-off

An inherent trade-off exists between privacy and model utility in differentially private machine learning. Stronger privacy guarantees usually come at the cost of reduced model accuracy. This trade-off can be controlled by adjusting the privacy parameters $\epsilon$ and $\delta$. When the parameters are chosen wisely, DP-SGD can achieve a level of privacy that is sufficient for many practical applications without sacrificing too much accuracy.

Researchers have proposed several techniques to mitigate the privacy-accuracy trade-off, such as privacy budget allocation, adaptive noise scaling, and differentially private data augmentation. These techniques aim to improve the model's utility while preserving the desired privacy guarantees.

Conclusion

Privacy is a crucial aspect of training machine learning models, as it helps to protect sensitive user information from being inadvertently leaked through the models. Differential privacy provides a rigorous framework for measuring privacy in machine learning and has been widely adopted in the field of privacy-preserving machine learning. Differentially private SGD is a promising approach that enables training machine learning models with strong privacy guaranteed by injecting noise into the gradients during optimization. However, it is essential to carefully balance the trade-off between privacy and model accuracy to achieve both private and useful models.