Gradient Descent's Sample Complexity in Stochastic Convex Optimization

Core Concepts

The sample complexity of full-batch Gradient Descent (GD) in stochastic convex optimization matches the worst-case sample complexity of empirical risk minimizers, showing no advantage over naive ERMs.

Abstract

The key highlights and insights from the content are:

The content analyzes the sample complexity of full-batch Gradient Descent (GD) in the setup of non-smooth Stochastic Convex Optimization (SCO).

It is shown that the generalization error of GD, with optimal choice of hyperparameters, can be Θ̃(d/m + 1/√m), where d is the dimension and m is the sample size. This matches the sample complexity of worst-case empirical risk minimizers.

The result implies that GD has no advantage over naive empirical risk minimization (ERM) algorithms in terms of sample complexity.

The analysis provides a new generalization bound that depends on both the dimension as well as the learning rate and number of iterations.

The bound also shows that, for general hyperparameters, when the dimension is strictly larger than the number of samples, T = Ω(1/ε^4) iterations are necessary to avoid overfitting. This resolves an open problem from prior work.

The content discusses the implications of the results, highlighting the importance of choosing the right algorithm for learning in SCO, as GD does not improve over the worst-case sample complexity of ERMs.

The paper also identifies open problems, such as the possibility of improved generalization bounds for GD in low dimensions, and the sample complexity of GD when the number of iterations is much larger than the sample size.

Stats

When d ≥ 4096, T ≥ 10, m ≥ 1, and η > 0, the generalization error of GD is lower bounded by Ω(min(d/(1032m), 1) * min(η/√(min{⌊d^3/136⌋,T}), 1)).
For d = Ω(m + T^(1/3)), the generalization error of GD is lower bounded by Ω(min(η/√T + 1/(ηT), 1)).
When T = O(m^1.5) and η = Θ(1/√T), the generalization error of GD is lower bounded by Ω(min(d/m + 1/√m, 1)).

Quotes

"The generalization error of GD, with (minmax) optimal choice of hyper-parameters, can be Θ̃(d/m + 1/√m), where d is the dimension and m is the sample size."
"This matches the sample complexity of worst-case empirical risk minimizers. That means that, in contrast with other algorithms, GD has no advantage over naive ERMs."
"Our bound also shows that, for general hyper-parameters, when the dimension is strictly larger than number of samples, T = Ω(1/ε^4) iterations are necessary to avoid overfitting."

Key Insights Distilled From

The Sample Complexity of Gradient Descent in Stochastic Convex Optimization

by Roi Livni at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04931.pdf

Deeper Inquiries

Can GD achieve a generalization error bound of O(dη/√T/m + 1/√m) in low dimensions, improving over the worst-case ERM bound

In low dimensions, Gradient Descent (GD) can achieve a generalization error bound of O(dη/√T/m + 1/√m), improving over the worst-case Empirical Risk Minimizer (ERM) bound. This improvement is significant as it shows that GD can generalize well even in scenarios where the dimension is relatively small compared to the number of samples. By carefully choosing the learning rate (η), number of iterations (T), and sample size (m), GD can outperform traditional ERM algorithms in terms of generalization error. This result highlights the effectiveness of GD in handling low-dimensional data and its ability to optimize the objective function efficiently.

What are the sample complexity implications if the number of GD iterations is much larger than the sample size, e.g., T = Ω(m^2)

If the number of GD iterations (T) is much larger than the sample size (m), for example, T = Ω(m^2), the sample complexity implications can be significant. In such a scenario, where the algorithm runs for a large number of iterations relative to the available samples, the risk of overfitting increases. This imbalance between the number of iterations and the sample size can lead to a higher risk of the algorithm memorizing the training data rather than generalizing well to unseen data. As a result, the algorithm may struggle to achieve good performance on new data points, leading to poor generalization and potentially higher prediction errors.

How does the sample complexity of GD compare to other algorithms, such as SGD or regularized ERMs, in the overparameterized regime where the dimension exceeds the number of samples

In the overparameterized regime where the dimension exceeds the number of samples, the sample complexity of Gradient Descent (GD) compared to other algorithms like Stochastic Gradient Descent (SGD) or regularized Empirical Risk Minimizers (ERMs) can vary. GD's sample complexity in this regime depends on factors such as the learning rate, number of iterations, and the dimension of the data. While GD can be effective in handling overparameterized scenarios, it may not always outperform other algorithms in terms of sample complexity. SGD, for example, is known for its efficiency in handling large datasets and high-dimensional data, while regularized ERMs offer a balance between model complexity and sample size, potentially leading to better generalization performance in certain cases. Therefore, the choice of algorithm in the overparameterized regime should be based on the specific characteristics of the data and the learning task at hand.