+

g
-

⁢
α

;

⁢

x
≤

-
α

⁢

ax
2

+
bx
+
c

;

⁢

-
α

≤
x
≤
β

g
+

⁢
x

+
t
+

α
+
β

2

⁢

g
-

+

α
-
β

2

⁢

g
+

;

⁢

x
>
β

⁢

The example embodiment described above, termed herein the Generalized Leaky “Smooth Rectified Linear Unit” or “SmeLU” activation function, illustrates an example of a larger family of activation functions that can be smooth, have a continuous gradient, and (optionally) be monotonic while mimicking the mathematical behavior of a Rectified Linear Unit (ReLU). Example variants within this larger family may have some or all of the following features.

As one example, in some implementations, the activation function can include a stop region having a gradient of about zero and a pass region with a constant positive gradient, similar to a ReLU function. For instance, the stop region and pass region can be defined by the linear piecewise segments. Thus, the activation function can achieve accuracy close to that of a ReLU function while having improved reproducibility due to satisfaction of constraints such as smoothness, continuous gradient, and/or monotonicity.

Additionally, the example embodiment described above can allow for variable values of the gradients (e.g., g+ and g−), vertical shift t, and coefficients α and β. For instance, in some implementations, the values of gradients and coefficients can be defined such that the activation function mimics the mathematical behavior of a ReLU activation function. For example, the gradients can be defined such that g− is zero and g+ is about one.

As another example, additionally or alternatively to the other example parameter values, the vertical shift can be defined as about zero. This allows for a full stop region at the left linear segment and a pass region with a gradient of about one at the right linear segment.

As another example, the coefficients can be defined such that α=β, achieving a “symmetric” activation function. In other words, the transition points can be symmetric about an origin (e.g., with transition region midpoint x=0). In some implementations, a simple symmetric version with a single parameter of SmeLU may be sufficient. Such an implementation can be obtained when α=β. Specifically, for the simple SmeLU,

$y = {\begin{matrix} 0; & x \leq - β \\ \frac{{(x + β)}^{2}}{4 β}; & \langle x \rangle \leq β \\ x; & x \geq β \end{matrix}$

As another example, a leaky piecewise activation function can include a g− that is greater than zero but still less than g+. In other words, the left linear segment can define a leaky region. Thus, the leaky piecewise activation function can mimic the mathematical behavior of a “leaky” ReLU activation function. The leaky piecewise activation function can achieve behavior closer to that of a linear activation function, which results in greater reproducibility. As a result, in some cases, the leaky piecewise activation function can achieve an improvement in reproducibility. An example of such an activation is given by

$y = {\begin{matrix} 0; & x \leq α \\ \frac{{(x + α)}^{2}}{2 (α + β)}; & - α \leq x \leq β \\ x + \frac{α - β}{2}; & x \geq β \end{matrix}$

In some implementations, a leaky version of simplified activation may be desirable. This can be achieved, for example, by g−>0, but g−<g+, with g+=1 and t=0.

As another example, the activation function can be shifted horizontally and/or vertically. For instance, the activation function can be shifted vertically by the vertical shift t and shifted horizontally by a horizontal shift s such that shifted activation z(x) equals original activation y(x) evaluated at y(x−s).

In some cases, the activation functions described herein can be used to approximate a generalized SoftPlus activation function, which is given as follows:

y=γ ln(1+exp(x/y)

The SoftPlus activation function provided above asymptotically approaches a vertically shifted symmetric version of one specific implementation of the proposed activation functions, approaching a gradient of zero farther left of the origin, and a gradient of one farther right from the origin. In the region x→0 from both sides, the Taylor series approximation

$y \approx γ \ln 2 + \frac{x}{2} + \frac{x^{2}}{8 γ}$

can be used to show that SoftPlus asymptotically approaches a vertically shifted up symmetric variant of the proposed SmeLU function, where β=2γ, and the vertical shift is a function of β, given by

$y_{softplus - γ} (x) = y_{β} (x) + \frac{β}{4} \ln \frac{4}{e}, β = 2 γ .$

Note, however, that the SoftPlus function does not provide a full stop in the stop region, especially for lower magnitude negative x, and larger values of γ.

As another example, the proposed activation function can be shifted such that the activation function crosses an origin. For instance, the activation function can output a zero if an input value is zero. The origin-crossing activation function can preserve a sign of the input value. In other words, the output can be negative and/or zero if the input is negative, and positive if the input is positive. One example origin-crossing piecewise activation function is given below.

$y = {\begin{matrix} g_{-} \cdot x + \frac{α^{2} (g_{-} - g_{+})}{2 (α + β)}; & x \leq - α \\ \frac{g_{+} - g_{-}}{2 (α + β)} x^{2} + \frac{α g_{+} + β g_{-}}{α + β} x; & - α \leq x \leq β \\ g_{+} \cdot x + \frac{β^{2} (g_{-} - g_{+})}{2 (α + β)}; & x \geq β \end{matrix}$

In some implementations, the activation function can include additional linear, polynomial (e.g., quadratic), and/or other non-linear piecewise segments. For instance, a smooth and continuous activation function with continuous gradient can include more than two linear segments, defining multiple leaky regions, full stop regions, and/or pass regions. For instance, one example embodiment includes a linear full stop region, a linear leaky region, and a linear pass region with quadratic transition regions between some or all of the linear regions. As another example, polynomial and/or other non-linear regions can be used in place of transition regions and/or linear regions.

In some implementations, values for the one or more activation function parameters can be learned by training. In some implementations, the values for the activation function parameters can be learned separately from the other parameters (e.g., weights, biases, etc.) of the network. Alternatively or additionally, the values for the activation function parameters can be learned jointly with the other parameters of the network. For instance, a neural network can be trained to learn values for the one or more activation function parameters that optimize for one or more training objectives, such as, but not limited to, accuracy, reproducibility, or any other suitable training objectives. For instance, optimizing for the one or more training objectives can include learning parameter values for the activation function parameters such that the one or more learning objectives are optimized (e.g., maximized and/or minimized) in objective space at the parameter values. In some cases, an overall training objective can be defined as a mean or weighted/costed sum of multiple training objectives. This can allow training to optimize for several training objectives, such as training objectives that may be conflicting (e.g., accuracy and reproducibility).

In some implementations, the activation function parameters can have identical values at each activation function in a neural network. Additionally or alternatively, respective values for the activation function parameters can be learned for each layer (e.g., hidden layer). For instance, the activation function parameters can have identical values for each activation function in one of the one or more layers, which may be different from values of activation function parameters in other layers. Additionally or alternatively, the activation function parameters can be learned uniquely for each activation node (e.g., each respective activation function implemented by each respective node).

In some implementations, the learning rate for learning the parameter values can be adjusted based on the choice of parameterization. Adjusting the learning rate can allow for better convergence (e.g., more accurate convergence) during training. For instance, in cases where the same parameter values are used at each activation function, the gradients can be summed from all units, whereas in cases where respective values for each activation function are learned uniquely, the gradients are sampled individually. Thus, the learning rate for the first case may need to be smaller than the second case to allow for better convergence.

Additionally, in some cases, the choice of parameterization can provide different results depending on the choice of training objective. For instance, if optimizing for accuracy, it may be desirable to learn parameter values for each layer or each activation function individually, as different layers can exhibit different trends of parameter values. For example, lower level layers may learn a negative g− while higher level layers may learn a monotonic parameterization.

In some cases, the neural network can be trained in multiple sessions. For example, in some implementations, the optimization of the activation function parameters is performed in an offline session. Thereafter, the learned activation function parameters are used wherein training the actual model in a subsequent session. Alternatively, the model (e.g., inclusive of activation function parameters) can be trained in a single session.

In some cases, reproducibility can be used as an explicit training objective (e.g., included and measured as part of the objective function). In one embodiment, reproducibility is included as a training objective during training of the model. Alternatively, however, the model is trained offline first to learn activation function parameters that are better for reproducibility, and then training is repeated using these activation function parameters to learn the model itself (e.g., with or without reproducibility as an explicit training objective). In some implementations, an ensemble can be used to optimize the activation function parameters offline (e.g., for reproducibility), and a single tower model can then be trained and deployed using the learned values of the activation function parameters(e.g., with or without reproducibility as an explicit training objective).

In some cases, ensembles can be used as a proxy for reproducibility during training. The ensembles may be in the same service, or across different services. For instance, in some cases, having the ensembles across different services may be more representative of deployment scenarios, and using the same service may be easier to implement. As one example, training can minimize a prediction difference loss or log-odds prediction difference between towers, or deep network components, of the ensemble. For instance, two towers can be trained to produce more identical predictions by imposing a loss on the deviation from one tower to the other tower. The loss can be propagated to the towers, and the networks in each tower move towards one another to improve the reproducibility between the predictions of the towers. While this can lead both towers to produce more similar predictions, it can undesirably reduce diversity provided by the different components of the ensemble.

Thus, it can be desirable to apply the cross-tower loss such that it does not diminish the diversity of the ensemble. One possible approach is to apply an L2 loss on the log-odds prediction difference between two towers and allow the gradients of this loss to propagate only to the learned activation function parameters of the piecewise activation function, and not to the actual model layer activations and the parameters that invoke them. For instance, a Stop-Gradient can be applied on the hidden layer nodes but not on the parameters for the L2 prediction difference loss. The model can be trained on a top level objective, where a Stop-Gradient is applied on the parameters but not on the activations in the hidden layers. This method can improve towards the objective with the learned values of the model parameters, while optimizing the learned values of the activation parameters to improve prediction difference. Other forms of losses can be used for the prediction difference objective, such as cross entropy loss that uses the prediction of one tower as the label for the other.

Another example advantage to training offline for reproducibility is that part of the benefit of the ensemble for reproducibility is attained by different initialization applied for the components of the ensemble. When the model is trained offline, towers that are initialized identically can be used without sacrificing the benefits of using an ensemble for training. In a first pass, the top level objective can be optimized for accuracy and the activation parameters can be optimized for reproducibility. In a second pass, the model can be trained with values of the activation function parameters fixed to the values learned in the first pass. If the deployed model is an ensemble, the components of the ensemble are now initialized differently such that the ensemble benefit can be maintained in the deployed model. If only a single tower is trained, the activation function parameters thus do not need to be optimized when training the model to be deployed.

In addition to learning parameters of the activations, the procedures described above can also be used to learn functional forms of the pieces of the smoothed piecewise activation. These can be learned while maintaining the continuity and smoothness (continuity of gradients) constraints. The number of pieces as well as the mathematical form of each piece can be learned, where the latter can be learned from a large given set of functional forms.

According to another aspect of the present disclosure, some of the proposed activation functions can provide the benefit of being deployable on simple hardware. For instance, some processing units (e.g., tensor processing deployment hardware units) may provide limited support for activation functions. For example, some tensor processing deployment hardware may only support ReLU and clipped linear or hard tan h activations. Thus, some activation functions, such as, for example, SoftPlus, GeLU, or SeLU, cannot be deployed on these tensor processing units.

In contrast, some implementations of the proposed activation functions can require only simple math to compute (e.g., first and second order polynomials can be used instead of relatively complicated functions such as exponential functions or higher-order polynomial functions).

Additionally, some implementations of the proposed activation functions can be expressed as a combination of shifted rectified linear unit functions and/or hard tan h functions. For example, a symmetric SmeLU activation function can be expressed according to the equations below. The equations given below can be implemented on simple hardware that only supports ReLU operations. Additionally, if the hardware supports hard tan h operations, it can also be used in place of the ReLU. Generally, fewer segments will provide better continuity of the gradient if the activation is deployed in training, although, usually, training has better functionality and can use the direct mathematical implementation. Deployment, on the other hand, may be limited.

ReLU implementations of the Symmetric can be:

$y = \frac{1}{4 β} [{ReLU}^{2} (x + β) - ReLU (x - β) ReLU (x) + β ReLU (x - β)]$

or the simpler form

$ReLUx (x, a) = \max {0, \min [x, a]}$

$y = \frac{1}{4 β} \cdot ReLUx (x + β, 2 * β) \cdot ReLU (x + β) + \frac{1}{2} \cdot ReLU (x - β)$

In some implementations, the activation function expressed as a piecewise function can be used for training and/or backpropagation, and the activation function expressed as a combination of ReLU and/or hard tan h functions can be used for deployment. Using the actual piecewise function for training can avoid influence from any potential gradient discontinuities and/or implementation constraints in the combination of functions with non-continuous gradients.

Activation functions according to example aspects of the present disclosure can achieve a number of technical effects and benefits. For example, activation functions according to example aspects of the present disclosure can be smooth. In other words, the activation functions can be continuous and have a continuous gradient. Additionally, the activation functions can be monotonic. Additionally, the activation functions can include any desirable type and number of regions, such as full stop regions, pass regions, and/or leaky regions. In this way, the activation functions according to example aspects of the present disclosure can achieve an improved tradeoff between accuracy and reproducibility, both for single tower and ensemble models, over existing activation functions such as ReLU while maintaining the behavior of the existing activation functions. Thus, multiple identically structured models can exhibit more consistent predictions while also providing desirable accuracy. Additionally, the activation functions according to example aspects of the present disclosure can be deployed on limited hardware that may not support more complex activation functions.

Example Neuron

FIG. 1 provides a graphical diagram of an example artificial neuron 10. The artificial neuron 10 can be connected to one or more presynaptic neurons 12, 14, 16. The artificial neuron 10 can be connected to the presynaptic neurons 12, 14, 16 via artificial synapses 18, 20, 22. The presynaptic neurons 12, 14, 16 can pass presynaptic neuron outputs to the artificial neuron 10 via the artificial synapses 18, 20, 22.

Each synapse 18, 20, 22 can have an adjustable weight 24, 26, 28 (e.g., scalar weight) associated therewith. The weights 24, 26, 28 can be changed as a result of learning. Each artificial synapse 18, 20, 22 can be either excitatory (e.g., have a positive-valued weight), which increases the summed input of the receiving neuron 10 upon receipt, or inhibitory (e.g., have a negative-valued weight), which decreases the summed input of the receiving neuron 10 upon receipt.

The artificial neuron 10 can also have an activation function 32, which controls the output 34 of the neuron 10 based on the summed input 30. In particular, the activation function 32 can be any of the proposed activation functions described herein (e.g., a smooth piece-wise continuous activation function). Use of an activation function 32 as described herein can improve reproducibility without sacrificing accuracy.

Although not explicitly shown in FIG. 1, various other parameters can impact the behavior of the artificial neuron 10 such as, for example, bias parameter(s), and/or the like.

Graphical Depictions of Example Activation Functions

FIG. 2 illustrates example activation functions according to example implementations of the present disclosure. In particular, curves 202-208 illustrate a family of piecewise activation functions having a left linear segment, middle quadratic segment, and right linear segment with varying parameter values. For instance, curves 202-208 illustrate a family of piecewise activation functions according to the equation below.

$y = {\begin{matrix} g_{-} x + t + g_{-} α; & x \leq - α \\ {ax}^{2} + bx + c; & - α \leq x \leq β \\ g_{+} x + t + \frac{α + β}{2} g_{-} + \frac{α - β}{2} g_{+}; & x > β \end{matrix}$

The particular example values for the parameters of the example activation functions of curves 202-208 are illustrated in FIG. 2.

FIG. 3 illustrates example activation functions according to example implementations of the present disclosure. In particular, curves 302-306 illustrate a family of “symmetric” smooth piecewise activation functions with varying parameter values. Specifically, curves 302-306 illustrate example activation functions according to the form above, where α=β, g−=0, g+=1, and t=0.

Example Devices and Systems

FIG. 4A depicts a block diagram of an example computing system 100 that can implement machine-learned models according to example implementations of the present disclosure. The system 100 includes a user computing device 102, a server computing system 130, and a training computing system 150 that are communicatively coupled over a network 180.

The user computing device 102 can be any type of computing device, such as, for example, a personal computing device (e.g., laptop or desktop), a mobile computing device (e.g., smartphone or tablet), a gaming console or controller, a wearable computing device, an embedded computing device, or any other type of computing device.

The user computing device 102 includes one or more processors 112 and a memory 114. The one or more processors 112 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, a FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 114 can include one or more non-transitory computer-readable storage mediums, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 114 can store data 116 and instructions 118 which are executed by the processor 112 to cause the user computing device 102 to perform operations.

In some implementations, the user computing device 102 can store or include one or more neural network models 120. For example, the neural network models 120 can be or can otherwise include various machine-learned models such as neural networks (e.g., deep neural networks) or other types of machine-learned models, including non-linear models and/or linear models. Neural networks can include feed-forward neural networks, recurrent neural networks (e.g., long short-term memory recurrent neural networks), convolutional neural networks or other forms of neural networks. Example neural network models 120 are discussed with reference to FIG. 2.

In some implementations, the one or more neural network models 120 can be received from the server computing system 130 over network 180, stored in the user computing device memory 114, and then used or otherwise implemented by the one or more processors 112. In some implementations, the user computing device 102 can implement multiple parallel instances of a single neural network model 120.

Additionally or alternatively, one or more neural network models 140 can be included in or otherwise stored and implemented by the server computing system 130 that communicates with the user computing device 102 according to a client-server relationship. For example, the neural network models 140 can be implemented by the server computing system 140 as a portion of a service (e.g., a web service). Thus, one or more models 120 can be stored and implemented at the user computing device 102 and/or one or more models 140 can be stored and implemented at the server computing system 130.

The user computing device 102 can also include one or more user input component 122 that receives user input. For example, the user input component 122 can be a touch-sensitive component (e.g., a touch-sensitive display screen or a touch pad) that is sensitive to the touch of a user input object (e.g., a finger or a stylus). The touch-sensitive component can serve to implement a virtual keyboard. Other example user input components include a microphone, a traditional keyboard, or other means by which a user can provide user input.

The server computing system 130 includes one or more processors 132 and a memory 134. The one or more processors 132 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, a FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 134 can include one or more non-transitory computer-readable storage mediums, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 134 can store data 136 and instructions 138 which are executed by the processor 132 to cause the server computing system 130 to perform operations.

In some implementations, the server computing system 130 includes or is otherwise implemented by one or more server computing devices. In instances in which the server computing system 130 includes plural server computing devices, such server computing devices can operate according to sequential computing architectures, parallel computing architectures, or some combination thereof.

As described above, the server computing system 130 can store or otherwise include one or more machine-learned neural network models 140. For example, the models 140 can be or can otherwise include various machine-learned models. Example machine-learned models include neural networks or other multi-layer non-linear models. Example neural networks include feed forward neural networks, deep neural networks, recurrent neural networks, and convolutional neural networks.

The user computing device 102 and/or the server computing system 130 can train the models 120 and/or 140 via interaction with the training computing system 150 that is communicatively coupled over the network 180. The training computing system 150 can be separate from the server computing system 130 or can be a portion of the server computing system 130.

The training computing system 150 includes one or more processors 152 and a memory 154. The one or more processors 152 can be any suitable processing device (e.g., a processor core, a microprocessor, an ASIC, a FPGA, a controller, a microcontroller, etc.) and can be one processor or a plurality of processors that are operatively connected. The memory 154 can include one or more non-transitory computer-readable storage mediums, such as RAM, ROM, EEPROM, EPROM, flash memory devices, magnetic disks, etc., and combinations thereof. The memory 154 can store data 156 and instructions 158 which are executed by the processor 152 to cause the training computing system 150 to perform operations. In some implementations, the training computing system 150 includes or is otherwise implemented by one or more server computing devices.

The training computing system 150 can include a model trainer 160 that trains the machine-learned models 120 and/or 140 stored at the user computing device 102 and/or the server computing system 130 using various training or learning techniques, such as, for example, backwards propagation of errors. For example, a loss function can be backpropagated through the model(s) to update one or more parameters of the model(s) (e.g., based on a gradient of the loss function). Various loss functions can be used such as mean squared error, likelihood loss, cross entropy loss, hinge loss, and/or various other loss functions. Gradient descent techniques can be used to iteratively update the parameters over a number of training iterations.

In some implementations, performing backwards propagation of errors can include performing truncated backpropagation through time. The model trainer 160 can perform a number of generalization techniques (e.g., weight decays, dropouts, etc.) to improve the generalization capability of the models being trained. In particular, the model trainer 160 can train the neural network models 120 and/or 140 based on a set of training data 162.

The model trainer 160 includes computer logic utilized to provide desired functionality. The model trainer 160 can be implemented in hardware, firmware, and/or software controlling a general purpose processor. For example, in some implementations, the model trainer 160 includes program files stored on a storage device, loaded into a memory and executed by one or more processors. In other implementations, the model trainer 160 includes one or more sets of computer-executable instructions that are stored in a tangible computer-readable storage medium such as RAM hard disk or optical or magnetic media.

The network 180 can be any type of communications network, such as a local area network (e.g., intranet), wide area network (e.g., Internet), or some combination thereof and can include any number of wired or wireless links. In general, communication over the network 180 can be carried via any type of wired and/or wireless connection, using a wide variety of communication protocols (e.g., TCP/IP, HTTP, SMTP, FTP), encodings or formats (e.g., HTML, XML), and/or protection schemes (e.g., VPN, secure HTTP, SSL).

The machine-learned models described in this specification may be used in a variety of tasks, applications, and/or use cases.

In some implementations, the input to the machine-learned model(s) of the present disclosure can be image data. The machine-learned model(s) can process the image data to generate an output. As an example, the machine-learned model(s) can process the image data to generate an image recognition output (e.g., a recognition of the image data, a latent embedding of the image data, an encoded representation of the image data, a hash of the image data, etc.). As another example, the machine-learned model(s) can process the image data to generate an image segmentation output. As another example, the machine-learned model(s) can process the image data to generate an image classification output. As another example, the machine-learned model(s) can process the image data to generate an image data modification output (e.g., an alteration of the image data, etc.). As another example, the machine-learned model(s) can process the image data to generate an encoded image data output (e.g., an encoded and/or compressed representation of the image data, etc.). As another example, the machine-learned model(s) can process the image data to generate an upscaled image data output. As another example, the machine-learned model(s) can process the image data to generate a prediction output.