The simplex method can outperform gradient descent when the loss function has many local minima.

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The question poses a claim about when the simplex method can outperform gradient descent. 
Option 1: True. Proponents might argue that the simplex method (e.g., the Nelder-Mead algorithm) is derivative-free and can explore the search space without relying on gradient information. In landscapes with many local minima, a gradient-based method can get trapped in a local basin, especially if gradients vanish or point toward suboptimal directions; a derivative-free approach might, in some cases, zigzag around and find a better basin. Additionally, simplex methods can adjust to irregular objective surfaces by contracting, expanding, or reflecting the simplex, which could occasionally help escape shallow local minima. However, this optimistic view has caveats: Nelder-Mead does not guarantee escaping all local minima and can deteriorate in high dimensions or with noisy evaluations. It is also not inherently designed to outperform gradient descent in the sense of always finding a global minimum, and performance is highly problem-dependent. 
Option 2: False. In many practical settings, the presence of numerous local minima does not guarantee that the simplex method will outperform gradient descent. Gradient descent with good scheduling, momentum, or initialization strategies, or with stochastic variants, can sometimes navigate complex landscapes more effectively than a pure Nelder-Mead approach. Moreover, simplex methods rely on function values and can be sensitive to scaling and dimensionality; they can get stuck in non-optimal basins or require many function evaluations to progress, which may be inefficient compared to well-tuned gradient-based methods. The claim generalizes performance in a way that contradicts the nuanced, problem-specific nature of optimization algorithms. 
Overall, while there are scenarios where a derivative-free simplex method may avoid some traps that gradient descent encounters, the assertion that it can outperform gradient descent specifically because the loss function has many local minima is not universally valid and is not a general rule. The statement is better understood as context-dependent and not universally true.

Learning AI Through Visualization 4 Module 2 Quiz

The simplex method can outperform gradient descent when the loss function has many local minima.

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