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## How do residual connections secretly struggle overfitting?

The thought in broad strokes is pretty easy: we will render weight decay virtually ineffective by making it arbitrarily small. Only a fast recap of what weight decay is: weight decay is a regularization method that’s used to forestall neural networks from converging to options that don’t generalize to unseen information (overfitting). If we prepare the neural community to solely decrease the loss on the coaching information we would discover a answer particularly tailor-made to this specific information and its idiosyncrasies. To keep away from that, we add a time period that corresponds to the norm of the burden matrices of the community. That is imagined to encourage the optimization course of to converge to options that may not be optimum for the coaching information, however have smaller weight matrices when it comes to norm. The pondering is that fashions which have excessive norm weights are much less pure and is likely to be making an attempt to suit particular information factors in an effort to decrease the loss a bit extra. In a means, it is a technique to combine Occam’s razor (the philosophical concept that easier options are most likely the suitable ones) into the loss — the place simplicity is captured by the norm of the weights. We won’t talk about deeper justifications for weight decay right here.

**TL;DR**:** **on this article,** **we present that in ReLU feedforward networks with LayerNorm that don’t have residual connections, the optimum loss worth is just not modified by weight decay regularization.

**Optimistic Scaling of ReLU Networks**

Each linear and ReLU networks share the next scaling characteristic: Let **a** > 0 be a optimistic scalar. Then **ReLU(ax) = a ReLU(x)**. Ultimately, in any community that’s composed of a stack of matrix multiplication adopted by a ReLU activation, this property will nonetheless maintain. That is probably the most vanilla type of neural community — no normalization layers and no residual connections. But, it’s moderately shocking that such feedforward (FF) networks which were ubiquitous not so way back display such a structured conduct: multiply your enter by a optimistic scalar and, lo and behold, the output is** **scaled **precisely** by the identical issue. That is what we name *(optimistic) *** scale equivariance** (that means that scaling within the enter interprets to scaling within the output, not like

*invariance*the place the output is just not affected in any respect by scaling within the enter)

*.*However there’s extra: if we do that with any of the burden matrices alongside the way in which (and the corresponding bias phrases) — the identical impact takes place: the output will probably be multiplied by the identical issue. Good? For positive. However can we use it? Let’s see.

Let’s see what occurs after we add LayerNorm. First, what’s LayerNorm? Fast recap:

the place **µ** is the typical of the entries of **x**, *◦*** **stands for elementwise multiplication, and **β, γ** are vectors.

So, what occurs to the scaling property after we add LayerNorm? After all, nothing adjustments earlier than the purpose we added the LayerNorm, so if we scaled a weight matrix by **a**>0 earlier than this level, the enter to the LayerNorm is scaled by **a**,** **after which what occurs is:

So, we get a brand new property, this time scaling simply leaves the output *unchanged — ***optimistic scale invariance**. And it’s about to remorse that…

** Observe: **whereas we talk about LayerNorm, different types of normalization, comparable to BatchNorm, fulfill the optimistic scale invariance property and so they’re as vulnerable as LayerNorm to the mentioned issues.

## Tips on how to Disappear Fully

Let’s remind ourselves what we’re making an attempt to reduce:

the place the coaching set is represented as a set of pairs of {** (x_i, y_i)**},

**and the parameters (weights) of the neural community**

**are designated by**

*f***Θ**. The expression is made from two components: the empirical loss to reduce — the lack of the neural community on the coaching set, and the regularization time period designed to make the mannequin attain “easier” options. On this case, simplicity is quantified because the weights of the community having low norms.

However right here’s the issue, we discovered a technique to bypass restrictions on the burden scale. We will scale each weight matrix by an arbitrarily small issue and nonetheless get the identical output. Stated in any other case — the **operate f**

*that each networks, the unique one and the scaled one, implement is strictly the identical! The internals would possibly differ, however the output is identical. It holds for each community with this structure, whatever the precise values of the parameters.*

Recall that generalization to unseen information is our aim. If the regularization time period goes to zero, the community is free to overfit the coaching information, and the regularization time period turns into ineffective. As now we have seen, for each community with such structure, we will design an equal community (i.e. computing precisely the identical operate) with arbitrarily small weight matrix norms, that means the regularization time period can go to zero with out affecting the empirical loss time period. In different phrases, we will take away the burden decay time period and it’s not going to matter.

A phrase of warning is due: whereas theoretically, the mannequin ought to discover a answer that overfits the coaching information, it has been noticed that optimization would possibly converge to generalizing options even with out express regularization. This has to do with the optimization algorithm. We use native optimization algorithms comparable to gradient descent, SGD, Adam, AdaGrad, and so forth. They aren’t assured to converge to probably the most ** globally** optimum answer. This generally occurs to be a blessing. An attention-grabbing line of labor (e.g., [Neyshabur, 2017]) means that these algorithms are a type of

**implicit regularization**, even when express regularization is lacking! It’s not bulletproof, however generally the mannequin converges to a generalizing answer — even with out regularization phrases!

Let me remind you what residual connections are. A residual connection provides the enter of the layer to the output. If the unique operate that the layer is computing is ** f(x) = ReLU(Wx)** then the brand new operate is

**.**

*x + f(x)*Now, the scaling property on weights breaks for this new layer. It’s because there is no such thing as a coefficient discovered in entrance of the residual a part of the expression. So the ** f(x)** half is being scaled by a continuing due to the burden scaling, however the

**half stays unchanged. Now, after we apply LayerNorm to this, the scaling issue can not cancel out:**

*x***.**

*LayerNorm(x + a f(x)) ≠ LayerNorm(x + f(x))***Importantly, it’s the case solely when the residual connection is utilized**

**LayerNorm. If we apply LayerNorm and solely then the residual connection, it seems that we nonetheless get the scaling invariance of LayerNorm:**

*earlier than***.**

*x + LayerNorm(a f(x)) = x + LayerNorm(f(x))*The primary variant is sometimes called the *pre-norm variant** *(extra exactly, it’s really ** x + f(LayerNorm(x)) **that known as this manner, however we will attribute the LayerNorm to the earlier layer, and take the subsequent layer’s LayerNorm

*yielding the above expression, aside for the sting instances of the primary and final layers). The second variant known as the*

**. These phrases are sometimes utilized in transformer architectures, that are out of the scope of this text. Nonetheless, it is likely to be attention-grabbing to say that a number of works comparable to [Xioang et al, 2020] discovered that pre-norm is simpler to optimize (they talk about totally different causes for the issue). Observe nevertheless this is probably not associated to the scaling invariance mentioned right here. Transformer pre-training datasets typically include big quantities of information, and overfitting turns into much less of an issue. Additionally, we haven’t mentioned transformer architectures per se. It’s nonetheless nonetheless one thing to consider.**

*post-norm variant*On this article, we noticed some attention-grabbing properties of feedforward neural networks with out pre-norm residual connections. Particularly, we noticed that if they do not include LayerNorm, they propagate enter scaling and weight scaling to the output. In the event that they do include LayerNorm, they’re scale-invariant, and weight/enter scaling doesn’t have an effect on the output in any respect. We used this property to point out that (arbitrarily near) the **optimum** options to such networks can keep away from any weight norm penalty, and so the community can converge to the identical answer it might have converged to with out them. Whereas it is a assertion about optimality, there’s nonetheless the query of whether or not these options are literally discovered utilizing gradient descent. We would sort out this in a future put up. We additionally mentioned how (pre-norm) residual connections break the dimensions invariance and thus appear to resolve the above theoretical downside. It’s nonetheless attainable that there will probably be related properties that residual connections couldn’t keep away from that I failed to contemplate. As all the time, I need to thanks for studying and I’ll see you within the subsequent put up!

F. Liu, X. Ren, Z. Zhang, X. Solar, and Y. Zou. *Rethinking residual reference to layer normalization,* 2020.

B. Neyshabur. *Implicit regularization in deep studying*, 2017. URL https://arxiv.org/abs/1709.01953.

R. Xiong, Y. Yang, D. He, Ok. Zheng, S. Zheng, C. Xing, H. Zhang, Y. Lan, L. Wang, and T.-Y. Liu. *On layer normalization within the transformer structure*, 2020. URL https://arxiv.org/abs/2002.04745.

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