How do GELU and SiLU treat a negative input differently from ReLU?
answer
- output equals input times a gate
- the gate is not just 0 or 1
- input times a probability
- Gaussian CDF for GELU, sigmoid for SiLU
- small negatives attenuated, not deleted
basics
~20 sReLU hard-zeroes every negative input. GELU and SiLU multiply the input by a smooth gate between 0 and 1 - a Gaussian CDF for GELU, a sigmoid for SiLU - so small negatives survive and the derivative stays continuous.
solid answer
~50 sAll three are self-gating: the output is the input times a gate. ReLU's gate is the indicator `1[x > 0]`, so it flips from 0 to 1 at the origin and every negative input becomes exactly 0. GELU's gate is `Phi(x)`, the standard normal CDF, so an input of -0.5 is multiplied by about 0.31 and comes out near -0.154 rather than being destroyed. SiLU (also called Swish) uses `sigmoid(x)` as the gate, giving about -0.189 at the same input; its gate closes more gradually, so it keeps more of a moderately negative value. For large positive inputs both gates approach 1 and the units behave like the identity, and for very negative inputs both approach 0. The whole difference lives in a band around the origin, and it costs an exponential or error function per element instead of a comparison.
go deeper
Be ready to state the two formulas plainly - input times the Gaussian CDF for GELU, input times the sigmoid for SiLU - and to say that a small negative input comes out small and negative rather than zero.
Explain the mechanics: the gate lives in [0, 1], equals 0.5 at the origin, tends to 1 for large inputs, and makes the derivative continuous instead of jumping. Be able to sketch both curves against ReLU.
Show judgment about when the swap is worth it: no parameter change, real per-element arithmetic cost, usually small accuracy movement, and always a retrain rather than a hot swap of the nonlinearity on fitted weights.
Own the standardisation call. Picking one default nonlinearity across an organisation's models buys shared kernels, comparable baselines and simpler deployment, and that consistency is often worth more than chasing per-model activation gains.
## What a hidden nonlinearity has to do Stack two linear layers with nothing between them and you get one linear map: depth buys you nothing. The elementwise nonlinearity between layers is what stops that collapse. Beyond that requirement, the design question is how the function should behave near zero, how it should behave far from zero, and what its derivative looks like - because that derivative is what backpropagation multiplies through the stack. ## All three units are gates A useful way to read this family is `output = input * gate(input)`, where the gate lands in [0, 1]: - **ReLU**: `max(0, x)` = `x * 1[x > 0]`. The gate is an indicator. It is 0 or 1, nothing between, and it switches discontinuously at the origin. - **GELU**: `x * Phi(x)`, where `Phi` is the cumulative distribution function of a standard normal - `Phi(x)` is the probability that a draw from a standard normal is at most `x`. The gate rises smoothly from 0 to 1, passing through 0.5 at `x = 0`. - **SiLU** (published as Swish): `x * sigmoid(x)` with `sigmoid(x) = 1 / (1 + exp(-x))`. Same shape of story, a different S-curve as the gate, also 0.5 at the origin. So GELU and SiLU are not "ReLU with the corner sanded off" in some vague sense; they are the same gating idea with a soft gate substituted for a hard one. ## Concrete numbers | x | ReLU | GELU | SiLU | |---|---|---|---| | -2.0 | 0 | -0.045 | -0.238 | | -0.5 | 0 | -0.154 | -0.189 | | 0.0 | 0 | 0 | 0 | | 0.5 | 0.5 | 0.346 | 0.311 | | 2.0 | 2.0 | 1.954 | 1.762 | Two things fall out. First, a moderately negative pre-activation is not annihilated - it is attenuated. Information about *how* negative it was still reaches the next layer. Second, SiLU's gate closes more slowly than GELU's, so SiLU keeps noticeably more of an input like -2 while GELU has already suppressed it to a few percent. ## The probabilistic reading of GELU GELU has a motivation beyond "a smooth curve that looked good". Suppose you decided to keep or zero each activation at random, with keep-probability equal to `Phi(x)` - that is, the more the input stands out against a standard normal, the more likely it survives. The expected value of that stochastic keep-or-zero is exactly `x * Phi(x)`. GELU is the deterministic expectation of an input-dependent random gate, which is why it is described as combining a nonlinearity with a stochastic-regularizer flavour in one function. ## Derivatives - ReLU: derivative is 0 for negative inputs and 1 for positive ones, with a jump at the origin (implementations pick a subgradient there). The kink is a single point, and gradient descent copes with it, but the derivative is not continuous. - GELU: `d/dx = Phi(x) + x * pdf(x)`, where `pdf` is the standard normal density. Continuous everywhere, equal to 0.5 at the origin. - SiLU: `d/dx = s(x) * (1 + x * (1 - s(x)))` with `s = sigmoid`. Also continuous everywhere, also 0.5 at the origin. The derivative of both smooth units exceeds 1 for a range of positive inputs before settling back toward 1, and both derivatives go negative in a small window on the negative side - which is another way of saying these functions are not monotone. ## The wider family of smooth ReLU approximations - **Softplus**: `log(1 + exp(x))`, a smooth, strictly positive approximation to ReLU. With a sharpness parameter it becomes `(1/beta) * log(1 + exp(beta * x))`, and as that parameter grows the curve sharpens back toward ReLU's kink - a knob that interpolates between smooth and hard. - **Mish**: `x * tanh(softplus(x))`, a smooth non-monotone unit built on top of softplus. - A tanh-based approximation of GELU exists, `0.5 * x * (1 + tanh(sqrt(2/pi) * (x + 0.044715 * x^3)))`, used when an error function is inconvenient or slow. ## Cost, and what it is not None of these units has parameters, so a swap changes no parameter count. What it changes is arithmetic per element: ReLU is a compare-and-select, while GELU and SiLU need an exponential or an error function for every activation value in every feature map. On large models with heavy matrix multiplications that is usually noise; on small networks with big feature maps it is not automatically noise, and should be measured rather than assumed. Finally: because the network's weights were fit against a particular nonlinearity, swapping the activation on an already-trained network and expecting the same accuracy is a mistake. Retrain, or at minimum fine-tune, after a swap.
- What does GELU output at exactly x = 0, and what is its gate value there?The gate is `Phi(0) = 0.5`, and the output is `0 * 0.5 = 0`. SiLU behaves the same way at the origin, since `sigmoid(0) = 0.5`. Both units therefore pass through the origin like ReLU, but with a derivative of 0.5 there instead of a jump between 0 and 1.
- Do GELU and SiLU differ meaningfully from ReLU for large positive inputs?No. Both gates approach 1 as the input grows, so the output approaches the input itself, exactly as ReLU does. The differences concentrate in a band around the origin, roughly where the absolute value of the input is below about 4. That is why these units are described as changing behaviour near zero, not far from it.
- What extra arithmetic does GELU cost per element compared with ReLU?ReLU is a comparison and a select. GELU needs an error function (or a tanh-based approximation of it), and SiLU needs an exponential, evaluated once per activation value. Neither adds parameters, so the extra cost is pure elementwise arithmetic over every value in every feature map.
ReLU is a light switch: off below zero, fully on above it. GELU and SiLU are dimmers whose brightness rises smoothly with the input, so a slightly negative signal comes through faintly instead of vanishing.
saying these in an interview costs you the question
- Says GELU is just ReLU and never outputs a negative value
- Claims the gate has learned parameters
- Treats SiLU as identical to a plain sigmoid activation
- Thinks the units differ from ReLU for large positive inputs
- Swaps activation on a trained network and expects unchanged accuracy