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Optimization Calculator

Calculate optimization convergence, computational cost, efficiency, and memory requirements for ML optimizers.

About this calculator

Training a machine learning model means repeatedly nudging its parameters in the direction that reduces error, and the choice of optimizer and learning rate strongly affects how fast that process converges and how much compute it costs. This calculator models convergence as geometric decay -- each iteration multiplies the remaining error by a convergence rate close to but below 1, so smaller rates mean faster convergence -- and estimates that rate from your learning rate and chosen optimizer: plain SGD scales directly with learning rate, while the adaptive optimizers Adam and RMSprop apply a steeper effective step (modeled here as 1.5x and 1.2x the learning rate respectively) to reflect their per- parameter step-size adaptation. From the convergence rate it estimates how many iterations are needed to reach a small error threshold, then reports Optimization Efficiency (how far your planned iteration count is from that estimate), Total Computational Cost (a rough forward-plus- backward-pass unit count scaled by parameter count and iterations), and Memory Required (accounting for the extra momentum/variance state that Adam and RMSprop store per parameter, unlike plain SGD).

This is a simplified illustrative model, not a rigorous convergence-theory proof -- real convergence behavior depends heavily on the objective function's actual curvature and conditioning. The Convex/Non-Convex selector approximates that dependency with a single multiplier: choosing Non-Convex halves each optimizer's effective per-step convergence progress for every optimizer, reflecting how saddle points and multiple local minima make practical progress toward a usable minimum less reliable than on a convex surface, where every step provably reduces distance to the single global minimum.

Inputs

Results

Convergence Rate

0.99

Expected Iterations

687

Optimization Efficiency

68.7%

Total Computational Cost20,000
Memory Required0.04 KB
How to Use This Calculator
  1. Select whether the Objective Function is Convex or Non-Convex and choose an Optimizer Type (SGD, Adam, or RMSprop).
  2. Enter Dimensions (number of parameters being optimized) and Learning Rate (optimization step size).
  3. Set Iterations to the number of optimization steps you plan to run.
  4. Review Convergence Rate and Expected Iterations to gauge how quickly the chosen optimizer should reach the convergence threshold.
  5. Check Total Computational Cost and Memory Required to estimate the resource footprint of training at this scale.

How the result changes with Learning Rate

Learning RateConvergence RateExpected IterationsOptimization Efficiency
0.010.9951,378137.8%
0.010.992591891.8%
0.020.98545745.7%
0.030.97527327.3%

What each input means

Objective Function
Type of objective function
Dimensions
Number of parameters to optimize
Learning Rate
Optimization step size
Iterations
Number of optimization iterations
Optimizer Type
Optimization algorithm

How this is calculated

Formula

θ_new = θ_old - α × ∇f(θ)

Worked example, using the default values

  1. Identify Input Parameters
    5 parameters
    Objective Function = 0, Dimensions = 10, Learning Rate = 0.01, Iterations = 1000, Optimizer Type = 0 = 5 input(s) provided
  2. Calculate Convergence Rate
    Convergence Rate
    0.99 = 0.99
  3. Calculate Expected Iterations
    Expected Iterations
    687 = 687
  4. Calculate Optimization Efficiency
    Optimization Efficiency
    68.7 = 68.7%
  5. Calculate Total Computational Cost
    Total Computational Cost
    20000 = 20000
  6. Calculate Memory Required
    Memory Required
    0.04 = 0.04

Engine last updated . Checked against 3 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.

Frequently Asked Questions

Why do Adam and RMSprop use a different convergence-rate formula than SGD?

Adam and RMSprop are adaptive optimizers -- they adjust the effective step size per parameter based on recent gradient history, which in practice tends to speed up early convergence compared to plain SGD's fixed step size. This calculator represents that with a steeper effective multiplier on learning rate (1.5x for Adam, 1.2x for RMSprop) so the convergence-rate estimate reflects their generally faster practical convergence.

What happens if I set the learning rate too high?

In real training, a learning rate that's too high for the chosen optimizer causes the loss to oscillate or diverge instead of converging smoothly -- especially with Adam or RMSprop's steeper effective step. This calculator's simplified model floors the convergence rate just above zero once the learning rate crosses that instability point, representing the fastest convergence its geometric-decay model can express rather than simulating literal divergence.

Why does memory requirement differ between SGD and the adaptive optimizers?

Adam and RMSprop both maintain extra per-parameter state between iterations -- running estimates of gradient momentum and/or squared-gradient variance -- so they need roughly twice the memory of plain SGD, which only tracks the parameters themselves. This overhead becomes significant at large parameter counts, which is one practical reason SGD or its momentum variants are still sometimes preferred for very large models.

Does Dimensions refer to the size of the input data or something else?

Dimensions here means the number of trainable parameters being optimized (the model's parameter count), not the size or dimensionality of the input data. It directly scales both the per-iteration computational cost and the memory required to store optimizer state, since more parameters means more values to update and track each step.

What actually changes when I switch Objective Function from Convex to Non-Convex?

Choosing Non-Convex halves the effective per-step convergence progress used in the convergence-rate formula for whichever optimizer you've selected, which raises the estimated Expected Iterations needed to converge and lowers Optimization Efficiency for the same planned iteration count. This models the real practical difference: gradient steps on a convex surface provably reduce distance to the single global minimum, while a non-convex surface's saddle points and multiple local minima make the same step size less reliable progress toward a usable result.

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