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Gradient Descent

Learn how gradient descent powers model training, from theory and variants to code and interview questions.

We'll cover the following...

How to explain gradient descent effectively
What are the different variants of gradient descent?
How to implement gradient descent
What about the learning rate?
Quiz
Conclusion

It’s hard to imagine a generative AI interview that doesn’t involve gradient descent. From simple linear regressions to cutting-edge neural networks, almost every model relies on some form of this algorithm to find the best parameters. Because it’s so foundational, many interviewers will question gradient descent to see if you truly grasp how models learn, beyond just memorizing training commands in a deep learning framework.

But how do you talk about gradient descent in an interview? You don’t just say, “It’s an optimization method.” Interviewers want to hear why this algorithm works. How do the different variants handle data differently? Why do you need a learning rate? How do you handle practical challenges like plateausPlateaus are flat regions in the loss surface where gradients vanish or become too small to drive meaningful updates. These often occur in deep networks due to poor initialization or vanishing gradients.?

We’ll cover each question in a structured way, highlighting the logic and potential follow-up questions, so you’ll be prepared for even the toughest “step-by-step” technical interviews.

How to explain gradient descent effectively

Gradient descent arose from the need to optimize model parameters by minimizing a loss function. In essence, we measure how “off” our predictions are (the loss) and adjust parameters (weights, biases, etc.) to reduce that loss. Mathematically, if $θ$ represents a set of parameters and $L(θ)$ is the loss, the update rule follows:

Where:

$α$ is the learning rate
$∇_{θ}L(θ)$ is the gradient of the loss concerning $θ$

This equation ensures that each parameter is nudged in the direction that most steeply reduces the current loss. Without this systematic step-by-step guide, our models would flail around guessing parameters arbitrarily or get stuck without a strategy for improvement.

Imagine hiking in a mountainous landscape at night with only a flashlight, where the ground under your feet represents your model’s loss. Each step is guided by a narrow beam of light that tells you whether the terrain (the gradient) slopes up or down. By always moving downward (the negative gradient), you gradually descend toward the valley, the lowest loss. Of course, you can overshoot if you take massive strides or inch along forever if you shuffle too slowly; that’s why the learning rate must balance speed with stability. The true genius of gradient descent is its reliance on local slope information to guarantee progress: you measure precisely how the ground tilts at your current position, then step against it. Over many small, consistent updates, these downhill moves accumulate into a reliable path to the valley floor—no guesswork required.

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In today’s generative AI systems, like large language models or text-to-image diffusion models, gradient descent powers the training processes. These models have millions (or even billions) of parameters, and each training iteration fine-tunes them to better predict the next token in a sentence or refine the details of a generated image. Advanced optimizers (e.g., Adam, RMSProp) build on gradient descent by adapting learning rates and incorporating momentum, but the underlying principle remains the same: calculate a gradient, move parameters downhill, and repeat. This iterative process, executed at a massive scale, allows GenAI systems to learn complex patterns and generate high-quality outputs.

What are the different variants of gradient descent?

Even though the fundamental principle of “moving downhill” remains the same, gradient descent can take on different forms depending on how much data is used to compute each update. Below are the three most common variants, each striking a different balance between computational cost and stability.

Batch gradient descent: The model processes the entire dataset before updating its parameters in this method. Because it accounts for every data point, batch gradient descent provides a highly accurate and stable gradient direction. However, it can become a major bottleneck when dealing with large datasets, as the algorithm must repeatedly iterate over the entire dataset to make a single update. Despite its computational expense, batch gradient descent often converges more predictably and is still favored in certain scenarios where data size is manageable and the problem demands maximum precision.
Stochastic gradient descent (SGD): On the opposite end of the spectrum, SGD uses just one training example to update parameters. This approach is lightning-fast per iteration, especially beneficial when datasets are huge, because it processes only a single data point before updating. The downside is that the gradient will naturally be noisier, which can make the training path jump around rather than smoothly descend. Interestingly, this “noise” can sometimes be advantageous, helping the model escape shallow local minima. However, the lack of stability means SGD may require additional techniques (like learning rate decay or momentum) for consistent convergence.
Mini-batch gradient descent: Sitting in the “just right” zone is mini-batch gradient descent. Instead of using the entire dataset or just one data point, you update parameters based on a small subset (or mini-batch) of the data at each step. This strategy balances between the speed of SGD and the stability of batch GD. By sampling batches randomly, you still get a good approximation of the overall loss while avoiding the computational overhead of processing every example each time. As a result, mini-batch gradient descent typically converges faster than batch gradient descent, with less variance in updates than pure SGD, making it the de facto choice for most deep learning pipelines today.

While mini-batch gradient descent often offers a balanced approach, each variant has strengths suited to different contexts. Batch gradient descent is ideal for small datasets or scenarios requiring stable, precise convergence, like scientific simulations. Stochastic gradient descent excels with streaming data or large datasets needing real-time updates. Depending on factors like latency, memory, or hardware, batch or stochastic methods may sometimes be the better choice.

Below is a concise comparison table for batch gradient descent, stochastic gradient descent, and mini-batch gradient descent:

Variant	Basic Approach	Pros	Cons	Typical Use Cases
Batch Gradient Descent	Uses the entire dataset to compute the gradient at each update.	- Very stable gradient estimate - Often converges smoothly	- Computationally expensive for large datasets - Requires significant memory	- Smaller datasets where full-batch processing is feasible - High-precision tasks (e.g., scientific simulations)
Stochastic Gradient Descent (SGD)	Uses one data point (or training example) at a time.	- Fast updates per iteration - Helps escape local minima due to noise	- Noisy gradient - Convergence can be less stable	- Streaming data - Extremely large datasets - Situations needing quick or online updates
Mini-Batch Gradient Descent	Uses a small subset (mini-batch) for each update.	- Balance between speed and stability - Good GPU utilization	- Requires tuning batch size - Some noise is still present, but less than pure SGD	- Deep learning pipelines - Common default in modern frameworks - Medium- to large-sized datasets

The table above highlights each method’s defining approach, key advantages, trade-offs, and typical usage scenarios. It equips you to make a more informed decision if given a scenario.

How to implement gradient descent

At this point, an interviewer might ask you to provide a code demo implementing one of the gradient descent variants. A classic starter example is being given a simple function like $f(x) = (x-2)^2$ , and asked to implement a version of a gradient descent approach to find the value of $x$ that minimizes $f(x)$ .

However, for technical or resource constraints, you can only call this function (and its gradient) once at a time, and you must immediately decide on the next $x$ value before calling again. You cannot store or accumulate multiple evaluations for a “batch” update. How would you iteratively update $x$ so that, after multiple steps, it converges near $x=2$ , the function’s global minimum? Typically, you’ll have:

initial_x (float)—your starting guess for x
learning_rate (float)—step size for each gradient update
num_iterations (int)—how many times to update x

The output will be the value of $x$ after performing this constrained gradient descent procedure. A frequent twist in interview scenarios requires you to implement the code without external libraries like scikit-learn or PyTorch, ensuring you understand the fundamental math and logic behind the updates.

Why don’t you go ahead and try it yourself? Your task is to use gradient descent to find the value of $x$ that minimizes the function. Fill in the missing parts below to compute and apply the gradient at each step.

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Python 3.10.4

# Gradient Descent - Starter Template
def optimize_single_query(initial_x, learning_rate, num_iterations):
    # Define the function f(x)
    def f(x):
        return (x - 2)**2
    # Define the gradient of f(x)
    def gradient(x):
        # TODO: Fill in the gradient of f(x)
        pass
    # Initialize x
    x = initial_x
    for i in range(num_iterations):
        # TODO: Compute the gradient at the current x
        grad = None
        # TODO: Update x using the gradient and learning rate
        x = None
        # Optional: print the current progress
        print(f"Iteration {i+1}: x = {x}, f(x) = {f(x)}")
    return x
# Example usage
initial_x = 10.0
learning_rate = 0.1
num_iterations = 20
final_x = optimize_single_query(initial_x, learning_rate, num_iterations)
print(f"Final value of x: {final_x:.4f}")

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Python 3.10.4

def optimize_single_query(initial_x, learning_rate, num_iterations):
    # f(x) = (x - 2)^2
    def f(x):
        return (x - 2)**2
    
    # Gradient f'(x) = 2 * (x - 2)
    def gradient(x):
        return 2*(x - 2)
    
    # Initialize x
    x = initial_x
    
    for i in range(num_iterations):
        # Single query: get gradient at the current x
        grad = gradient(x)
        
        # Immediately update x in the negative gradient direction
        x = x - learning_rate * grad
        
        # Print progress (optional for debugging)
        print(f"Iteration {i+1}: x = {x:.4f}, f(x) = {f(x):.4f}")
    
    return x
# Example usage
initial_x = 10.0
learning_rate = 0.1
num_iterations = 20
final_x = optimize_single_query(initial_x, learning_rate, num_iterations)
print(f"Final value of x: {final_x:.4f}")

In the code above:

Lines 4–9: We define our function f(x) = (x - 2)^2 and its derivative gradient(x) = 2*(x - 2). These are essential for calculating the loss (the function value) and how it changes concerning x (the gradient).
Line 12: We initialize our parameter x with the user-provided initial_x. This gives us a starting point before any updates are performed.
Lines 14–22: We enter a loop that runs for num_iterations. In each iteration, we:
- Compute the gradient at the current x.
- Update x by moving in the negative direction of the gradient, scaled by learning_rate.
- Print the iteration number, current x, and f(x) to observe the convergence process.
Line 24: After the loop finishes, we return the final value of x. By this point, we hope it has moved close to x = 2, the global minimum of our function.
Lines 27–32: This is the example usage. We provide an initial_x, learning_rate, and num_iterations, call the function, and then print out the result. In a typical interview or testing environment, you might tweak these arguments to see how starting position or learning rate changes affect convergence.

From the output, you can see that each iteration moves $x$ closer to $2$ , causing $f(x) = (x-2)^2$ to decrease steadily. Initially, $x$ is far from $2$ , so the function value is large. As you step through each iteration, the gradient updates pull $x$ in the negative direction of the slope, lowering the loss at each stage. Notice how $f(x)$ drops more dramatically in the early iterations and then settles into smaller and smaller improvements as $x$ nears the optimum. By iteration 20, $x$ has converged to around $2.09$ , and $f(x)$ has shrunk to $0.0085$ , which is quite close to the global minimum at $x = 2$ .

As we have only one function value/gradient available at a time (i.e., one “data point”), each update is done immediately after a single gradient evaluation—mirroring SGD. In a scenario with multiple data points, batch gradient descent would process all data simultaneously, whereas mini-batch would process a small subset. Here, you’re forced into the single-sample approach by the constraint that you cannot “batch” multiple evaluations together.

Now, suppose you have multiple data points—for example, an array of inputs $X = \bigl[x_1,\, x_2,\, \dots,\, x_n\bigr]$ . Instead of a single function $(x - 2)^2$ , you might have a loss that sums or averages over these data points. In other words, something like:

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Python 3.10.4

import numpy as np
def optimize_batch_query(X, w_init, learning_rate, num_iterations):
    # Suppose we define: f(w) = sum((w - x_i)^2 for x_i in X)
    def f(w, X):
        return np.sum((w - X)**2)
    # gradient(w) = 2 * sum(w - x_i for x_i in X)
    def gradient(w, X):
        return 2 * np.sum(w - X)
 
    w = w_init
    for i in range(num_iterations):
        # Batch: compute gradient across all data points in X
        grad = gradient(w, X)
        w = w - learning_rate * grad
        print(f"Iteration {i+1}: w = {w:.4f}, f(w) = {f(w, X):.4f}")
 
    return w
    
X_data = np.array([1.0, 2.0, 3.0, 4.0], dtype=float)
        
# Set hyperparameters
w_init = 0.0
learning_rate = 0.01
num_iterations = 10
        
# Run batch gradient descent
w_final = optimize_batch_query(X_data, w_init, learning_rate, num_iterations)
print(f"\nFinal value of w: {w_final:.4f}")

In the above code:

Line 3: We define the function optimize_batch_query, which takes:
- X (a NumPy array of data points),
- w_init (initial guess for the parameter w),
- learning_rate (the size of each gradient descent step), and
- num_iterations (how many times we update w).
Lines 6–7: We define the function f(w, X)
Lines 10–11: The actual implementation of the gradient function, which again uses NumPy’s vectorized operations to sum the differences w - X, then multiply by 2.
Line 13: We set our parameter w to the initial guess w_init.
Line 14: We enter a for loop, running num_iterations times.
Line 16: We calculate the batch gradient by calling grad = gradient(w, X). This uses all data points in X.
Line 17: We update w in the negative direction of the gradient, multiplied by the learning_rate. This is the core update rule for gradient descent:
Line 20: We return the final value of w after num_iterations updates.

By the end, w should trend toward reducing the sum of squares, ideally moving near the mean of the dataset if the learning rate and number of iterations are chosen appropriately.

What about the learning rate?

When performing gradient descent, the learning rate ( $\alpha$ ) controls how big a step you take in the opposite direction of the gradient. If it’s too large, your parameter updates may overshoot the minimum and cause the loss to bounce around or even diverge. If it’s too small, you’ll make snail-like progress, potentially getting stuck or taking an extremely long time to reach a good solution. In short:

Overshooting (learning rate too high) occurs when the training process “jumps around” without settling, sometimes getting worse instead of better after each step.
Slow convergence (learning rate too low) occurs when the model improves gradually, so reaching the minimum might take an impractically large number of iterations.

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Striking a balance is essential. Often, people use a learning rate scheduler (e.g., reducing every few epochs) to start with bigger steps for rapid initial descent and then fine-tune with smaller steps as the model nears a minimum. In practice—especially with deep neural networks—you may also encounter situations where the loss function plateaus early and stops decreasing. This can happen for many reasons apart from the learning rate, and there’s no one-size-fits-all fix. Here are some strategies you might consider to solve the issues:

Advanced optimizers like Adam dynamically adapt learning rates for each parameter. They’re better at navigating plateaus and can speed up convergence, especially in very deep or complex networks.
Deep networks sometimes suffer from gradients becoming extremely small as you backprop through many layers. Techniques like skip connections A technique that allows a layer to “skip” one or more layers and feed its output directly into a later layer. Commonly used in ResNets to combat vanishing gradients. (ResNets), carefully chosen activation functions (e.g., ReLU variants), or simply reducing network depth can alleviate this problem.
Traditional SGD can stall if the gradient is very small or the terrain is flat. Adding momentum helps carry the model parameters through areas of low gradient, like pushing a heavy ball across a nearly level plane.

A rare follow-up: What is SGD with momentum and why is it commonly used?

In traditional SGD, parameters are updated strictly in the negative gradient direction of the loss function, which can be slow or oscillatory when the loss surface has many ridges and valleys. SGD with momentum tackles this by adding a “memory” of previous gradients, effectively giving each update an extra push in the same direction. You can think of it like rolling a heavy ball downhill: once it gains speed in one direction, it’s harder to stop.

Mathematically, the algorithm maintains a velocity variable, which is a blend of the current gradient and the past update. This velocity is then added to the parameter update, allowing for faster movement through shallow regions and reducing erratic oscillations. The momentum factor often between 0.8 and 0.99 determines how much of the previous update’s direction is carried forward, and the learning rate still governs overall step size. By smoothing out the trajectory in the parameter space, momentum helps the model converge more quickly and avoid getting stuck in small, unhelpful minima.

Ultimately, handling plateaus is about diagnosing why the model has stopped improving—be it a suboptimal learning rate, a challenging loss landscape, or network design issues—and then applying a targeted fix.

Quiz

Here's a quick question for you:

Suppose you are training a large language model with a dataset that is orders of magnitude larger than memory, on highly parallelized hardware (such as GPUs/TPUs). Your goal is to maximize convergence speed without sacrificing too much stability, while also efficiently utilizing hardware and minimizing communication overhead between processing units. Which variant of gradient descent and configuration is most likely to deliver optimal training performance in this scenario, and why?

A. Batch gradient descent with infrequent but highly accurate updates, because it guarantees the lowest-variance parameter updates and thus ensures the most stable convergence path.

B. Stochastic gradient descent with learning rate decay and momentum, because it provides the fastest per-iteration updates and leverages the noise to avoid local minima, requiring only a single sample per update.

C. Stochastic gradient descent with batch normalization on a single data point per update, to reduce update variance and speed up training for extremely large models.

D. Mini-batch gradient descent with moderate batch sizes (e.g., 128–1024), because it enables parallel computation of gradients across batches, balances variance and convergence stability, and aligns well with the memory and communication constraints of modern hardware.

Conclusion

Mastering gradient descent is a foundational milestone on your path to proficiency in artificial intelligence—whether you’re building models or training massive generative AI architectures.

Once you’ve demonstrated a solid grasp of gradient descent basics, interviewers may follow up with more advanced questions to assess your deeper understanding. These might include:

“How does Adam differ from standard SGD?”
“Why might learning rate decay help with convergence?”
“How would you tune batch size in a resource-constrained environment?”

Being prepared to answer these shows not just that you know how gradient descent works, but that you understand when and why to adapt it—skills that are essential when training real-world models.

Introduction

Neural Network Training and Optimization

Embeddings and Tokenization

Attention Mechanisms

Evaluation Techniques

Model Architectures and Comparisons

Learning Techniques

Scalability and Efficiency

Wrap Up

Fundamentals of Generative AI

Gradient Descent

How to explain gradient descent effectively

What are the different variants of gradient descent?

How to implement gradient descent

What about the learning rate?

Quiz

Conclusion