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According to these assumptions, except for ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:a(n.msub,{children:[e(n.mi,{children:"s"}),e(n.mn,{children:"0"})]})}),e(n.annotation,{encoding:"application/x-tex",children:"s_0"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".5806em",verticalAlign:"-.15em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",children:"s"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3011em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"0",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing 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",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:a(n.msub,{children:[e(n.mi,{children:"s"}),a(n.mrow,{children:[e(n.mi,{mathvariant:"script",children:"P"}),e(n.mo,{stretchy:"false",children:"("}),e(n.mi,{children:"k"}),e(n.mo,{stretchy:"false",children:")"})]})]})}),e(n.annotation,{encoding:"application/x-tex",children:"s_{\\mathcal{P}(k)}"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".7858em",verticalAlign:"-.3552em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",children:"s"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t 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The two viewpoints are essentially equivalent. The process of executing a program is described in Algorithm 1."]}),a(n.h4,{className:"content-header",id:"handling-multiple-program-inputs-or-outputs",children:[e(n.a,{href:"#handling-multiple-program-inputs-or-outputs","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),e(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"Handling Multiple Program Inputs or Outputs"]}),a(n.p,{children:["When a program has multiple inputs, we always group them 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Two possible representations of a program. ",e(n.strong,{children:"Left:"})," Functions are represented as nodes and dependencies as edges. ",e(n.strong,{children:"Right:"})," Functions are represented as nodes and dependencies as a set of disjoint nodes. In both cases, the input is represented by a root node and the output by a leaf node."]})}),a(n.h4,{className:"content-header",id:"alternative-program-representation-bipartite-graphs",children:[e(n.a,{href:"#alternative-program-representation-bipartite-graphs","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),e(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"Alternative Program Representation: Bipartite Graphs"]}),a(n.p,{children:["In our formalism, since a function ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:a(n.msub,{children:[e(n.mi,{children:"f"}),e(n.mi,{children:"k"})]})}),e(n.annotation,{encoding:"application/x-tex",children:"f_k"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".8889em",verticalAlign:"-.1944em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".10764em"},children:"f"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t 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",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:a(n.msub,{children:[e(n.mi,{children:"f"}),e(n.mi,{children:"k"})]})}),e(n.annotation,{encoding:"application/x-tex",children:"f_k"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".8889em",verticalAlign:"-.1944em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".10764em"},children:"f"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3361em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"-.1076em",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing 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However, it is also possible to consider two disjoint sets of nodes: one for functions and another for outputs, i.e., a bipartite graph. This formalism is similar to factor graphs (",e(n.em,{children:"Frey et al"}),"., ",e(n.a,{href:"https://people.ee.ethz.ch/~loeliger/localpapers/FG_Allerton1997.pdf",children:"1997"}),"; ",e(n.em,{children:"Loeliger"}),", ",e(n.a,{href:"https://ieeexplore.ieee.org/document/1267047",children:"2004"}),") used in probabilistic modeling, but with directed edges. An advantage of this formalism is that it allows functions to have multiple outputs. For simplicity, we will focus on the representation of a single set of nodes."]}),a(n.h3,{className:"content-header",id:"arithmetic-circuits",children:[e(n.a,{href:"#arithmetic-circuits","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),e(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"Arithmetic Circuits"]}),a(n.p,{children:["Arithmetic circuits are one of the simplest examples of computational graphs, originating from ",e(n.strong,{children:"computational complexity theory"}),". Formally, an arithmetic circuit over a field ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{mathvariant:"double-struck",children:"F"})}),e(n.annotation,{encoding:"application/x-tex",children:"\\mathbb{F}"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6889em"}}),e(n.span,{className:"mord mathbb",children:"F"})]})})]}),", such as the real numbers ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{mathvariant:"double-struck",children:"R"})}),e(n.annotation,{encoding:"application/x-tex",children:"\\mathbb{R}"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6889em"}}),e(n.span,{className:"mord mathbb",children:"R"})]})})]})," or complex numbers ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{mathvariant:"double-struck",children:"C"})}),e(n.annotation,{encoding:"application/x-tex",children:"\\mathbb{C}"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6889em"}}),e(n.span,{className:"mord mathbb",children:"C"})]})})]}),", is a directed acyclic graph (DAG) where the root nodes are elements of the field ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{mathvariant:"double-struck",children:"F"})}),e(n.annotation,{encoding:"application/x-tex",children:"\\mathbb{F}"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6889em"}}),e(n.span,{className:"mord mathbb",children:"F"})]})})]})," and the intermediate functions are arithmetic operations such as addition or multiplication. The latter are usually called ",e(n.strong,{children:"arithmetic gates"}),". Contrary to the general case of computational graphs, each function, whether a sum or a product, can have multiple root nodes. Root nodes can be variables or constants and must belong to the field ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{mathvariant:"double-struck",children:"F"})}),e(n.annotation,{encoding:"application/x-tex",children:"\\mathbb{F}"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6889em"}}),e(n.span,{className:"mord mathbb",children:"F"})]})})]}),". Arithmetic circuits can be used to calculate polynomials. There can be multiple arithmetic circuits to represent a given polynomial. To compare two arithmetic circuits that represent the same polynomial, an intuitive notion of ",e(n.strong,{children:"complexity"})," is the ",e(n.strong,{children:"circuit size"}),", as defined below."]}),e(m,{title:"Definition 1. 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For example, given a dataset of pairs ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[a(n.mrow,{children:[e(n.mo,{stretchy:"false",children:"("}),a(n.msub,{children:[e(n.mi,{children:"x"}),e(n.mi,{children:"i"})]}),e(n.mo,{separator:"true",children:","}),a(n.msub,{children:[e(n.mi,{children:"y"}),e(n.mi,{children:"i"})]}),e(n.mo,{stretchy:"false",children:")"})]}),e(n.annotation,{encoding:"application/x-tex",children:"(x_i, y_i)"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:"1em",verticalAlign:"-.25em"}}),e(n.span,{className:"mopen",children:"("}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",children:"x"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3117em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"0",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:e(n.span,{className:"mord mtight mathnormal",children:"i"})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".15em"},children:e(n.span,{})})})]})})]}),e(n.span,{className:"mpunct",children:","}),e(n.span,{className:"mspace",style:{marginRight:".1667em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".03588em"},children:"y"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3117em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"-.0359em",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:e(n.span,{className:"mord mtight mathnormal",children:"i"})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".15em"},children:e(n.span,{})})})]})})]}),e(n.span,{className:"mclose",children:")"})]})})]}),", we can minimize the loss function ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[a(n.mrow,{children:[e(n.mi,{children:"L"}),e(n.mo,{stretchy:"false",children:"("}),e(n.mi,{children:"w"}),e(n.mo,{stretchy:"false",children:")"}),e(n.mo,{children:"="}),a(n.msubsup,{children:[e(n.mo,{children:"∑"}),a(n.mrow,{children:[e(n.mi,{children:"i"}),e(n.mo,{children:"="}),e(n.mn,{children:"1"})]}),e(n.mi,{children:"m"})]}),e(n.mi,{mathvariant:"normal",children:"ℓ"}),e(n.mo,{stretchy:"false",children:"("}),e(n.mi,{children:"f"}),e(n.mo,{stretchy:"false",children:"("}),a(n.msub,{children:[e(n.mi,{children:"x"}),e(n.mi,{children:"i"})]}),e(n.mo,{separator:"true",children:";"}),e(n.mi,{children:"w"}),e(n.mo,{stretchy:"false",children:")"}),e(n.mo,{separator:"true",children:","}),a(n.msub,{children:[e(n.mi,{children:"y"}),e(n.mi,{children:"i"})]}),e(n.mo,{stretchy:"false",children:")"})]}),e(n.annotation,{encoding:"application/x-tex",children:"L(w) = \\sum_{i=1}^{m} \\ell(f(x_i; 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The minimization of the loss function can be done using an optimization algorithm such as gradient descent."]}),a(n.h2,{className:"content-header",id:"multilayer-perceptrons",children:[e(n.a,{href:"#multilayer-perceptrons","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),e(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"Multilayer Perceptrons"]}),a(n.h3,{className:"content-header",id:"combination-of-affine-layers-and-activation-functions",children:[e(n.a,{href:"#combination-of-affine-layers-and-activation-functions","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),e(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"Combination of Affine Layers and Activation Functions"]}),e(n.p,{children:"In the previous section, we did not specify how to parameterize feedforward networks. 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In this case, the activation function ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:a(n.msub,{children:[e(n.mi,{children:"a"}),e(n.mn,{children:"1"})]})}),e(n.annotation,{encoding:"application/x-tex",children:"a_1"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".5806em",verticalAlign:"-.15em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",children:"a"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3011em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"0",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:e(n.span,{className:"mord mtight",children:"1"})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".15em"},children:e(n.span,{})})})]})})]})]})})]})," is the identity function. 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This is the main attraction of MLPs: the ability to learn intermediate representations that are useful for the task of interest. We will see that MLPs can also be used as subcomponents in other architectures."]}),a(n.h2,{className:"content-header",id:"activation-functions",children:[e(n.a,{href:"#activation-functions","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),e(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"Activation Functions"]}),a(n.p,{children:["As mentioned earlier, feedforward networks use activation functions ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:a(n.msub,{children:[e(n.mi,{children:"a"}),e(n.mi,{children:"k"})]})}),e(n.annotation,{encoding:"application/x-tex",children:"a_k"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".5806em",verticalAlign:"-.15em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",children:"a"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3361em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"0",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:e(n.span,{className:"mord mtight mathnormal",style:{marginRight:".03148em"},children:"k"})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".15em"},children:e(n.span,{})})})]})})]})]})})]})," for each layer. In this section, we will describe some of the most common scalar-to-scalar or vector-to-scalar activation functions. We will also present some probability functions that can be used as activation functions."]}),a(n.h3,{className:"content-header",id:"nonlinear-scalar-to-scalar-activation-functions",children:[e(n.a,{href:"#nonlinear-scalar-to-scalar-activation-functions","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),e(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"Nonlinear Scalar-to-Scalar Activation Functions"]}),a(n.p,{children:["The ",e(n.strong,{children:"ReLU"})," (Rectified Linear Unit) function is a nonlinear activation function defined as the non-negative part of its argument. That is, the ReLU function is the identity function for non-negative values and zero for negative 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A multilayer perceptron with ReLU activations is called a rectifier neural network. The layers take the 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Furthermore, the logistic function can be obtained as the derivative of the ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[a(n.mrow,{children:[e(n.mtext,{children:"softplus"}),e(n.mo,{stretchy:"false",children:"("}),e(n.mi,{children:"x"}),e(n.mo,{stretchy:"false",children:")"})]}),e(n.annotation,{encoding:"application/x-tex",children:"\\text{softplus}(x)"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:"1em",verticalAlign:"-.25em"}}),e(n.span,{className:"mord text",children:e(n.span,{className:"mord",children:"softplus"})}),e(n.span,{className:"mopen",children:"("}),e(n.span,{className:"mord mathnormal",children:"x"}),e(n.span,{className:"mclose",children:")"})]})})]})," function, that is, for all 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In this case, the intermediate functions are parameterized functions that are applied sequentially to an input. Indeed, consider a feedforward network with ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[a(n.mrow,{children:[e(n.mi,{children:"n"}),e(n.mo,{children:"+"}),e(n.mn,{children:"1"})]}),e(n.annotation,{encoding:"application/x-tex",children:"n+1"})]})})}),a(n.span,{className:"katex-html","aria-hidden":"true",children:[a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6667em",verticalAlign:"-.0833em"}}),e(n.span,{className:"mord mathnormal",children:"n"}),e(n.span,{className:"mspace",style:{marginRight:".2222em"}}),e(n.span,{className:"mbin",children:"+"}),e(n.span,{className:"mspace",style:{marginRight:".2222em"}})]}),a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6444em"}}),e(n.span,{className:"mord",children:"1"})]})]})]})," layers ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[a(n.mrow,{children:[a(n.msub,{children:[e(n.mi,{children:"f"}),e(n.mn,{children:"1"})]}),e(n.mo,{separator:"true",children:","}),e(n.mo,{children:"…"}),e(n.mo,{separator:"true",children:","}),a(n.msub,{children:[e(n.mi,{children:"f"}),a(n.mrow,{children:[e(n.mi,{children:"n"}),e(n.mo,{children:"+"}),e(n.mn,{children:"1"})]})]})]}),e(n.annotation,{encoding:"application/x-tex",children:"f_1, \\ldots, f_{n+1}"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".9028em",verticalAlign:"-.2083em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".10764em"},children:"f"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3011em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"-.1076em",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:e(n.span,{className:"mord mtight",children:"1"})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".15em"},children:e(n.span,{})})})]})})]}),e(n.span,{className:"mpunct",children:","}),e(n.span,{className:"mspace",style:{marginRight:".1667em"}}),e(n.span,{className:"minner",children:"…"}),e(n.span,{className:"mspace",style:{marginRight:".1667em"}}),e(n.span,{className:"mpunct",children:","}),e(n.span,{className:"mspace",style:{marginRight:".1667em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".10764em"},children:"f"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3011em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"-.1076em",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:a(n.span,{className:"mord mtight",children:[e(n.span,{className:"mord mtight mathnormal",children:"n"}),e(n.span,{className:"mtight mbin",children:"+"}),e(n.span,{className:"mord mtight",children:"1"})]})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".2083em"},children:e(n.span,{})})})]})})]})]})})]}),". Surely, whenever ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:a(n.msub,{children:[e(n.mi,{children:"f"}),a(n.mrow,{children:[e(n.mi,{children:"n"}),e(n.mo,{children:"+"}),e(n.mn,{children:"1"})]})]})}),e(n.annotation,{encoding:"application/x-tex",children:"f_{n+1}"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".9028em",verticalAlign:"-.2083em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".10764em"},children:"f"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3011em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"-.1076em",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:a(n.span,{className:"mord mtight",children:[e(n.span,{className:"mord mtight mathnormal",children:"n"}),e(n.span,{className:"mtight mbin",children:"+"}),e(n.span,{className:"mord mtight",children:"1"})]})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".2083em"},children:e(n.span,{})})})]})})]})]})})]})," is the identity function, the set of functions that this network can represent is the same as that of an ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"n"})}),e(n.annotation,{encoding:"application/x-tex",children:"n"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".4306em"}}),e(n.span,{className:"mord mathnormal",children:"n"})]})})]}),"-layer network ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[a(n.mrow,{children:[a(n.msub,{children:[e(n.mi,{children:"f"}),e(n.mn,{children:"1"})]}),e(n.mo,{separator:"true",children:","}),e(n.mo,{children:"…"}),e(n.mo,{separator:"true",children:","}),a(n.msub,{children:[e(n.mi,{children:"f"}),e(n.mi,{children:"n"})]})]}),e(n.annotation,{encoding:"application/x-tex",children:"f_1, \\ldots, f_n"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".8889em",verticalAlign:"-.1944em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".10764em"},children:"f"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3011em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"-.1076em",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:e(n.span,{className:"mord mtight",children:"1"})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".15em"},children:e(n.span,{})})})]})})]}),e(n.span,{className:"mpunct",children:","}),e(n.span,{className:"mspace",style:{marginRight:".1667em"}}),e(n.span,{className:"minner",children:"…"}),e(n.span,{className:"mspace",style:{marginRight:".1667em"}}),e(n.span,{className:"mpunct",children:","}),e(n.span,{className:"mspace",style:{marginRight:".1667em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".10764em"},children:"f"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".1514em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"-.1076em",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:e(n.span,{className:"mord mtight mathnormal",children:"n"})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".15em"},children:e(n.span,{})})})]})})]})]})})]}),". Therefore, we can consider an ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[a(n.mrow,{children:[e(n.mi,{children:"n"}),e(n.mo,{children:"+"}),e(n.mn,{children:"1"})]}),e(n.annotation,{encoding:"application/x-tex",children:"n+1"})]})})}),a(n.span,{className:"katex-html","aria-hidden":"true",children:[a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6667em",verticalAlign:"-.0833em"}}),e(n.span,{className:"mord mathnormal",children:"n"}),e(n.span,{className:"mspace",style:{marginRight:".2222em"}}),e(n.span,{className:"mbin",children:"+"}),e(n.span,{className:"mspace",style:{marginRight:".2222em"}})]}),a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6444em"}}),e(n.span,{className:"mord",children:"1"})]})]})]}),"-layer network as an ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"n"})}),e(n.annotation,{encoding:"application/x-tex",children:"n"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".4306em"}}),e(n.span,{className:"mord mathnormal",children:"n"})]})})]}),"-layer network. In other words, depth, in theory, should not harm the expressive power of feedforward networks. Unfortunately, the assumption that each ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:a(n.msub,{children:[e(n.mi,{children:"f"}),e(n.mi,{children:"k"})]})}),e(n.annotation,{encoding:"application/x-tex",children:"f_k"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".8889em",verticalAlign:"-.1944em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".10764em"},children:"f"}),e(n.span,{className:"msupsub",children:a(n.span,{className:"vlist-t vlist-t2",children:[a(n.span,{className:"vlist-r",children:[e(n.span,{className:"vlist",style:{height:".3361em"},children:a(n.span,{style:{top:"-2.55em",marginLeft:"-.1076em",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:e(n.span,{className:"mord mtight mathnormal",style:{marginRight:".03148em"},children:"k"})})]})}),e(n.span,{className:"vlist-s",children:""})]}),e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".15em"},children:e(n.span,{})})})]})})]})]})})]})," is an identity function is not realistic. This means that deeper networks can sometimes be more difficult to train than shallower ones, causing accuracy to saturate or degrade as a function of depth."]}),a(n.p,{children:["The key idea of residual neural networks (ResNets) (",e(n.em,{children:"He et al."}),", ",e(n.a,{href:"https://arxiv.org/pdf/1512.03385",children:"2016"}),") is to introduce ",e(n.strong,{children:"residual blocks"})," between layers, such that the output of one layer becomes the input to a later layer. Formally, a residual block is a function of the 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It's important to note that the parameters of ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"g"})}),e(n.annotation,{encoding:"application/x-tex",children:"g"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".625em",verticalAlign:"-.1944em"}}),e(n.span,{className:"mord mathnormal",style:{marginRight:".03588em"},children:"g"})]})})]})," and ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"h"})}),e(n.annotation,{encoding:"application/x-tex",children:"h"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6944em"}}),e(n.span,{className:"mord mathnormal",children:"h"})]})})]})," are shared across time steps. Typically, ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"g"})}),e(n.annotation,{encoding:"application/x-tex",children:"g"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".625em",verticalAlign:"-.1944em"}}),e(n.span,{className:"mord mathnormal",style:{marginRight:".03588em"},children:"g"})]})})]})," and ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"h"})}),e(n.annotation,{encoding:"application/x-tex",children:"h"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6944em"}}),e(n.span,{className:"mord mathnormal",children:"h"})]})})]})," are parameterized using single hidden layer MLPs. Note that ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"g"})}),e(n.annotation,{encoding:"application/x-tex",children:"g"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".625em",verticalAlign:"-.1944em"}}),e(n.span,{className:"mord mathnormal",style:{marginRight:".03588em"},children:"g"})]})})]})," has multiple inputs."]}),a(n.h3,{className:"content-header",id:"sequence-to-vector",children:[e(n.a,{href:"#sequence-to-vector","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 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Many to one (encoding). 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Note that we denote the length of the target sequence as ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:a(n.msup,{children:[e(n.mi,{children:"l"}),e(n.mo,{lspace:"0em",mathvariant:"normal",rspace:"0em",children:"′"})]})}),e(n.annotation,{encoding:"application/x-tex",children:"l'"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".7519em"}}),a(n.span,{className:"mord",children:[e(n.span,{className:"mord mathnormal",style:{marginRight:".01968em"},children:"l"}),e(n.span,{className:"msupsub",children:e(n.span,{className:"vlist-t",children:e(n.span,{className:"vlist-r",children:e(n.span,{className:"vlist",style:{height:".7519em"},children:a(n.span,{style:{top:"-3.063em",marginRight:".05em"},children:[e(n.span,{className:"pstrut",style:{height:"2.7em"}}),e(n.span,{className:"mtight sizing reset-size6 size3",children:e(n.span,{className:"mord mtight",children:e(n.span,{className:"mord mtight",children:"′"})})})]})})})})})]})]})})]})," instead of ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"l"})}),e(n.annotation,{encoding:"application/x-tex",children:"l"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6944em"}}),e(n.span,{className:"mord mathnormal",style:{marginRight:".01968em"},children:"l"})]})})]})," to avoid confusion. However, in practice, the length may depend on the input and is often not known in advance. To address this issue, the vocabulary (of size ",a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"d"})}),e(n.annotation,{encoding:"application/x-tex",children:"d"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".6944em"}}),e(n.span,{className:"mord mathnormal",children:"d"})]})})]}),' in our notation) is typically augmented with an "end of sequence" (EOS) token, so that at inference time, we know when to stop generating the output sequence. A disadvantage of this architecture is that all information about the input sequence is contained in the context vector ',a(n.span,{className:"katex",children:[e(n.span,{className:"katex-mathml",children:e(n.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:a(n.semantics,{children:[e(n.mrow,{children:e(n.mi,{children:"c"})}),e(n.annotation,{encoding:"application/x-tex",children:"c"})]})})}),e(n.span,{className:"katex-html","aria-hidden":"true",children:a(n.span,{className:"base",children:[e(n.span,{className:"strut",style:{height:".4306em"}}),e(n.span,{className:"mord mathnormal",children:"c"})]})})]}),", which can therefore become a bottleneck."]}),a(n.h2,{className:"content-header",id:"conclusions",children:[e(n.a,{href:"#conclusions","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),e(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"Conclusions"]}),e(n.p,{children:"Neural networks, conceived as parameterized programs, provide a powerful and flexible framework for addressing a wide range of problems in machine learning and data processing. Throughout this article, we have explored various architectures and configurations, from feedforward networks to recurrent neural networks and residual networks. The representation of these architectures as directed acyclic graphs (DAGs) not only facilitates their analysis and optimization but also allows for the efficient application of techniques such as automatic differentiation. The different configurations discussed demonstrate the versatility of these models for tasks ranging from image captioning to machine translation."}),e(n.p,{children:"This approach of viewing neural networks as parameterized programs not only allows for designing more efficient and effective models but also opens pathways for improving the interpretability and explainability of these models, an area of growing importance in artificial intelligence. Although we have covered several types of neural networks, it's important to note that the field is constantly evolving, with new architectures and techniques being developed continuously. In summary, this perspective provides a solid foundation for the future development of more advanced, interpretable, and adaptable machine learning models for a wide range of real-world applications."}),a(n.p,{children:["Finally, if there are any errors, omissions, or inaccuracies in this article, please don't hesitate to contact us through the following Discord channel: ",e(n.a,{href:"https://discord.gg/gJ3vCgSWeh",children:"Math & Code"}),"."]}),a(n.h2,{className:"content-header",id:"references",children:[e(n.a,{href:"#references","aria-hidden":"true",tabIndex:"-1",children:e(s,{children:e(n.span,{className:"content-header-link",children:a(n.svg,{viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",className:"h-5 linkicon w-5",fill:"currentColor",children:[e(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),e(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"References"]}),e(r,{type:"image",url:"https://unsplash.com/es/fotos/una-imagen-generada-por-computadora-de-un-cerebro-humano-woGRo9m-ByE",text:"Randa Marzouk - Unplash"}),e(r,{type:"arxiv",url:"https://arxiv.org/pdf/2403.14606",text:"Mathieu Blondel and Vincent Roulet. 2024. The Elements of Differentiable Programming."})]})}return{default:function(s={}){const{wrapper:a}=s.components||{};return a?e(a,{...s,children:e(_createMdxContent,{...s})}):_createMdxContent(s)}};function _missingMdxReference(s,e){throw new Error("Expected "+(e?"component":"object")+" `"+s+"` to be defined: you likely forgot to import, pass, or provide it.")}

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