Chat GPT: Calculate Parameters in Model

ChatGPT 4

You

I'm training a feedforward neur network model. It's input has 1536 dimensions (a dense feature vector). Then there's 2048 size layer followed by 512 weights one and finally a single output with sigmoid activation. Could you help me to calculate the number of parameters in the model? (Do not hesitate to use interpreter to perform calculations)

ChatGPT

python
# Calculating the number of parameters in the given feedforward neural network model

# Input layer to first hidden layer parameters
input_to_first_hidden = 1536 * 2048

# First hidden layer biases
first_hidden_biases = 2048

# First hidden layer to second hidden layer parameters
first_to_second_hidden = 2048 * 512

# Second hidden layer biases
second_hidden_biases = 512

# Second hidden layer to output layer parameters
second_to_output = 512 * 1

# Output layer biases
output_biases = 1

# Total number of parameters
total_parameters = input_to_first_hidden + first_hidden_biases + first_to_second_hidden + second_hidden_biases + second_to_output + output_biases

total_parameters

Result

The feedforward neural network model you described has a total of 4,197,377 parameters.

You

OK. Let's say I'd like to scale the input vector length N and use bigger features but keep the overall model number of parameters largely the same. Could you help me create appropriate formula for that and calculate the hidden layers size for 4096 length input. Let's try to reserve the magnitude of the ratio between hidden layers (it's 4:1 so it should not fall below 3:1 and grow above 6:1). What sizes of input vectors would meet that requirement?

ChatGPT

To scale the input vector length $N$ and adjust the hidden layer sizes while keeping the overall number of parameters the same, we can start by formulating the total number of parameters in terms of $N$ , the size of the first hidden layer $H_1$ , and the size of the second hidden layer $H_2$ . Given the constraint on the ratio between the hidden layers ( $3:1 \leq \frac{H_1}{H_2} \leq 6:1$ ), we'll also incorporate this into our calculation.

The total number of parameters $P$ in your model is given by the sum of:

The weights between the input layer and the first hidden layer: $N \times H_1$
The biases for the first hidden layer: $H_1$
The weights between the first hidden layer and the second hidden layer: $H_1 \times H_2$
The biases for the second hidden layer: $H_2$
The weights between the second hidden layer and the output layer: $H_2 \times 1$
The biases for the output layer: $1$

This gives us the equation: $P = N \times H_1 + H_1 + H_1 \times H_2 + H_2 + H_2 + 1$

Given $P = 4,197,377$ (from the initial model) and a new input vector length $N = 4096$ , we want to find $H_1$ and $H_2$ that satisfy the ratio constraints and the total parameter count.

Let's simplify the problem by assuming the biases contribute relatively little to the total parameter count and can be approximately neglected in the calculations for $H_1$ and $H_2$ . This simplification allows us to focus on finding $H_1$ and $H_2$ more directly.

We can start by solving the equation for $H_1$ and $H_2$ given $N = 4096$ and $P = 4,197,377$ , keeping in mind the ratio constraints.

python
from sympy import symbols, Eq, solve

# Define symbols
N, H1, H2, P = symbols('N H1 H2 P')

# Total number of parameters (approximated, excluding biases)
P = 4197377  # Total parameters from the previous model
N_new = 4096  # New input vector length

# Equation representing the total number of parameters excluding biases for simplicity
parameters_eq = Eq(N_new * H1 + H1 * H2 + H2, P)

# We also need to consider the ratio constraints for H1 and H2: 3:1 <= H1/H2 <= 6:1
# However, directly incorporating these constraints into symbolic solving might be complex,
# so we'll first solve for H1 in terms of H2, then find a suitable H2 that fits the constraints.

# Solve for H1 in terms of H2
H1_solution = solve(parameters_eq, H1)[0]

# Now, we need to find H2 values that fit the ratio constraints and then calculate H1
# Instead of solving it symbolically, which might be complex due to the constraints,
# we'll approach this by iterating over possible H2 values within a reasonable range and checking the constraints.

# This is a heuristic approach to find a suitable H2 value that fits the constraints and total parameters.
# The optimal approach would involve a more detailed search or optimization algorithm,
# but this should give us a reasonable estimate.

# Placeholder for the solution
H2_solution = None
H1_final = None

# Iterate over a range of H2 values to find a suitable one
for possible_H2 in range(1, 1000):  # Arbitrary range, can be adjusted based on expected sizes
    possible_H1 = H1_solution.subs(H2, possible_H2)
    ratio = possible_H1 / possible_H2
    if 3 <= ratio <= 6 and N_new * possible_H1 + possible_H1 * possible_H2 + possible_H2 <= P:
        H2_solution = possible_H2
        H1_final = possible_H1
        break

H1_final, H2_solution

Result

(4197212/4261, 165)

To maintain the overall number of parameters largely the same for an input vector length of 4096, while also respecting the constraint that the ratio between the sizes of the first and second hidden layers should not fall below 3:1 and not grow above 6:1, we can use the following sizes for the hidden layers:

The size of the first hidden layer ( $H_1$ ) can be approximately $4197212/4261$ , which simplifies to around 985.
The size of the second hidden layer ( $H_2$ ) would be 165.

These values are calculated to keep the total number of parameters close to the original model's total parameter count and maintain a ratio within the specified range. Note that due to the simplification (ignoring biases in the calculation) and the discrete nature of the sizes (since layer sizes are integers), these values are approximations aimed at meeting the specified criteria.

You

Let's assume there are no biases at all. What is the number of parameters in the original network with 1536 input and 2048, 512 hidden layers and single outpu?

ChatGPT

python
# Recalculating the number of parameters in the original network without considering biases

# Input layer to first hidden layer parameters without biases
input_to_first_hidden_no_biases = 1536 * 2048

# First hidden layer to second hidden layer parameters without biases
first_to_second_hidden_no_biases = 2048 * 512

# Second hidden layer to output layer parameters without biases
second_to_output_no_biases = 512 * 1

# Total number of parameters without biases
total_parameters_no_biases = input_to_first_hidden_no_biases + first_to_second_hidden_no_biases + second_to_output_no_biases

total_parameters_no_biases

Result

Without considering biases, the original network with an input of 1536 dimensions, hidden layers of sizes 2048 and 512, and a single output, has a total of 4,194,816 parameters.

You

Wat is the number of weights in the 4096 input network with 985 and 165 hidden layer usind the same assumptions?

ChatGPT

python
# Calculating the number of parameters in the network with 4096 input, 985 and 165 hidden layers, without considering biases

# Input layer to first hidden layer parameters without biases for the new network
input_to_first_hidden_no_biases_new = 4096 * 985

# First hidden layer to second hidden layer parameters without biases for the new network
first_to_second_hidden_no_biases_new = 985 * 165

# Second hidden layer to output layer parameters without biases for the new network
second_to_output_no_biases_new = 165 * 1

# Total number of parameters without biases for the new network
total_parameters_no_biases_new = input_to_first_hidden_no_biases_new + first_to_second_hidden_no_biases_new + second_to_output_no_biases_new

total_parameters_no_biases_new

Result

The network with an input of 4096 dimensions, hidden layers of sizes 985 and 165, and a single output, assuming no biases, has a total of 4,197,250 parameters.

You

the ratio between hidden layers is now close to 6:1, could you try to bring it closer to 4:1?

ChatGPT

python
# To adjust the ratio between the hidden layers closer to 4:1 for the network with 4096 input,
# we need to find new sizes for H1 and H2 that also maintain the total number of parameters close to the original value.
# The target is to adjust H1 and H2 such that H1/H2 ≈ 4

# Given the total number of parameters without biases is approximately 4,197,250 (from the previous calculation),
# and the new input vector length is 4096, we aim to adjust H1 and H2 to achieve a 4:1 ratio.

# Let's define a target ratio
target_ratio = 4

# To find H1 and H2 that satisfy this ratio and the total parameter count,
# we can use an iterative approach to adjust H2 and calculate H1 based on the target ratio and total parameters.

# Find a suitable H2 value that fits the new ratio constraint and then calculate H1
H2_adjusted_solution = None
H1_adjusted_final = None

# Recalculate total parameters to consider
P_adjusted = 4197250  # Using the previously calculated total parameters as a reference

# Iterate over a range of H2 values to adjust the ratio to 4:1
for possible_H2 in range(1, 1000):  # Adjust the range based on expected sizes
    possible_H1 = target_ratio * possible_H2
    total_params_current = N_new * possible_H1 + possible_H1 * possible_H2 + possible_H2
    if abs(total_params_current - P_adjusted) < P_adjusted * 0.01:  # Allowing some deviation in total parameters
        H2_adjusted_solution = possible_H2
        H1_adjusted_final = possible_H1
        break

H1_adjusted_final, H2_adjusted_solution

Result

(960, 240)

To bring the ratio between the hidden layers closer to 4:1 for the network with a 4096 input vector, adjusting the sizes to 960 for the first hidden layer and 240 for the second hidden layer would be a suitable approach. This configuration maintains the total number of parameters close to the original value while achieving a ratio that is approximately 4:1.

You

What is the total number or weights now?

ChatGPT

python
# Calculating the number of parameters in the adjusted network with 4096 input, 960 and 240 hidden layers, without considering biases

# Input layer to first hidden layer parameters without biases for the adjusted network
input_to_first_hidden_no_biases_adjusted = 4096 * 960

# First hidden layer to second hidden layer parameters without biases for the adjusted network
first_to_second_hidden_no_biases_adjusted = 960 * 240

# Second hidden layer to output layer parameters without biases for the adjusted network
second_to_output_no_biases_adjusted = 240 * 1

# Total number of parameters without biases for the adjusted network
total_parameters_no_biases_adjusted = input_to_first_hidden_no_biases_adjusted + first_to_second_hidden_no_biases_adjusted + second_to_output_no_biases_adjusted

total_parameters_no_biases_adjusted

Result

The adjusted network with an input of 4096 dimensions, hidden layers of sizes 960 and 240, and a single output, without considering biases, has a total of 4,162,800 parameters.