Source: Deep Learning on Medium
Bye to Trial and Error Activation Functions of Neural Networks III: Proposed Jameel’s ANNAF Deterministic Criterion
“You never change things by fighting the existing reality. To change something, build a new model that makes the existing model obsolete.”
“Your brain does not manufacture thoughts. Your thoughts shape neural networks.”
In my third article entitled “Bye to Trial and Error Activation Functions of Neural Networks II: Proposed Jameel’s ANNAF Stochastic Criterion”, a stochastic criterion was discussed. It would be recalled that eight (8) stocks prices activation functions were proposed. In this article, using “Jameel’s ANNAF Deterministic Criterion”, we can have up to at least TWO THOUSAND (2000) ACTIVATION FUNCTIONS emanated from our sample AI-ML-PURIFIED DATA SET.
Proposed Jameel’s ANNAF Deterministic Criterion
ANNAF means Artificial Neural Network Activation Functions.
For any Neural Network that require DETERMINISTIC ACTIVATION FUNCTIONS can satisfy the following proposed criterion:
(1) The function f(x)shall be EMANATED from our referenced AI-ML-Purified Data Set. The essence of the function f(x) to be EMANATED from the referenced AI-ML-Purified Data is to build an incredible and sophisticated Activation Function(s) that has the BEST MATCH AND OR TUNE with our referenced AI-ML-Purified Data Set since the neural network is a system made to learn a function from data. The Activation Functions obtained from the referenced AI-ML-Purified Data can be used to build an extra-ordinary Neural Network Artificial Intelligence System.
(2) A curve fitting for Best Fitted Deterministic Function shall be carried out, the function f(x)whose:
(a) Rank is Unity (1)
(b) Fattiness Standard Error is smaller than any other on the list;
(3) The function f(x) shall be Nonlinear;
(4)The function f(x) shall have a Range;
(5) The function f(x) shall be Continuously Differentiable;
(6) The function f(x) shall be Monotonic;
(7) The function f(x) shall be Smooth Function with a Monotonic Derivative;
(8) The function f(x) shall Approximates Identity near the Origin.
If these failed Discard the 1st rated function f(x), repeat (1) to (8) until the qualified Deterministic Activation Function is EMANATED from our referenced AI-ML-Purified Data.
NOTE: Deep Learning Artificial Neural Network’s Hidden and output Layers consist of at least one, two or more Best fitted Activation Functions EMANATED from our AI-Data Set, therefore, the RANK: UNITY (ONE) in (2)-(a) and Fattiness Standard Error (2)-(b) of the criterion means when a function whose Real “Rank =1” was chosen and it satisfied (1) to (8) then the next function on list whose Real “Rank=2” will assume “New Rank=1” and will be tested to satisfy all the eight (8) axioms until we have the required number of BEST (EXCELLENT) Activation Functions needed to carry out our Deep Learning Artificial Neural Network.
Proposed Jameel’s Deterministic Lemma
All the TOP-RANKED Nonlinear Monotonic Continuously Differentiable Deterministic Functions EMANATED from referenced AI-ML-Purified Data satisfies Proposed “Jameel’s ANNAF Deterministic Criterion” are EXCELLENT DETERMINISTIC ACTIVATION FUNCTIONS to perform well-informed Forward and Backward Propagations of an Artificial Neural Network.
Jamilu (2019) employed the “TABLECURVE 2D CURVE FITTING” software of “SYSTAT”. The software automatically fits 3,665 BUILT-IN EQUATIONS FROM ALL DISCIPLINES to discover the ideal MODELS that describe Data. “TableCurve 2D is the first and only program that completely eliminates endless “TRIAL and ERROR” by automating curve fitting process”. This is because the software statistically RANKED the LIST of candidate equations according to “Jameel’s ANNAF Deterministic Criterion” of Deep Learning Artificial Neural networks proposed above.
The author used the SAMPLE DATA of “TEMPERATURE VS CONDUCTANCE” provided by TABLECURVE 2D Software as shown below:
Deterministic Activation Functions for Temperature vs Conductance
Now let find the BEST fitting Functions of the above AI DATA SET:
The FIRST Ranked Function is:
We can view all the FUNCTIONS that fitted our Sample AI DATA as follows:
Automatically fitting and added 2224 Functions at the end of 1:45 minutes as follows:
The SECOND Ranked Function is:
The Rank of about 2153 fitted Functions Emanated from our Sample AI Data:
This means we can have up to about 2224 FUNCTIONS (mostly Deterministic) that can be served as ACTIVATION FUNCTIONS EMANATED from our SAMPLE AI DATA to perform Deep Learning Neural Network for “TEMPERATURE vs CONDUCTANCE”. Thus, these satisfied first axiom of “Jameel’s ANNAF Deterministic Criterion” that says “(1) The function shall be EMANATED from the referenced AI-ML-Purified Data Set”.
Now we will test and see whether all the 2224 set of Activation Functions satisfies the remaining SEVEN (7) AXIOMS of “Jameel’s ANNAF Deterministic Criterion “ for the successful conduct of our Deep Learning Neural Network processes. Note that any qualified Stochastic Activation Function in this list shall satisfy “Jameel’s ANNAF Stochastic Criterion”.
The author only shows the First Derivatives of the THREE TOP-RANKED Activation Functions as follows:
Backward Propagation: First Derivatives of the Three Top-Ranked Activation Functions
First Derivative of the First (1ST) Ranked Activation Function:
First Derivative of the Second (2ND) Ranked Activation Function:
Now if a=b=c=d=e=f=g=h=i=j=k=1 then we have:
First Derivative of the Third (3ND) Ranked Activation Function:
Now if a=b=c=d=e=f=g=h=i=j=1 then we have:
Thanks to “TABLECURVE 2D CURVE FITTING” software of “SYSTAT” and “DERIVATIVE CALCULATOR”, the Three Top-Ranked Activation Functions are now DIFFERENTIABLE.
Thus, satisfied the following AXIOMS of “Jameel’s ANNAF Deterministic Criterion”:
(1) The THREE (3) functions f(x)s EMANATED from our referenced (Temperature vs Conductance) AI-ML-PURIFIED DATA SET.
(2) The THREE (3) functions f(x)s’ curve fittings have:
(a) Rank = 1
(b) Fattiness Standard Error is smaller than any others on the list;
(3) The functions f(x)s are Nonlinear;
(4)The functions f(x)s all have Ranges;
(5) The functions f(x)s are Continuously Differentiable;
(6) The functions f(x)s are Monotonic;
(7) The functions f(x)s are Smooth Function with a Monotonic Derivative;
And so on.
Therefore, if the Three Top-Ranked Activation Functions satisfied the remaining axiom of “Jameel’s ANNAF Deterministic Criterion”, are ready for the successful conduct of Deep learning Neural Network for “TEMPERATURE vs CONDUCTANCE” otherwise, the iterations shall be repeated until all the Activation Functions satisfies the Deterministic Criterion.
Guarantee, if the first three deterministic functions satisfied Jameel’s ANNAF Deterministic Criterion are excellent Activation Functions to successfully conduct “TEMPERATURE vs CONDUCTANCE” Deep Learning Neural Network. Subsequent Functions on the list are also good Advanced Optimized Activation Functions.
This research REVEALED that the Advanced Activation Functions satisfies Jameel’s ANNAF Stochastic and or Deterministic Criterion would henceforth depend on the REFERENCED PURIFIED AI DATA SET, TIME CHANGE and AREA OF APPLICATION (acronym DTA) as shown in the figure below:
The direction of my next article would work towards achieving “SUPER-INTELLIGENT NEURAL NETWORKS” using “Jameel’s ANNAF Stochastic Criterion” and “Jameel’s ANNAF Deterministic Criterion”.
Jamilu Auwalu Adamu, FIMC, CMC, FIMS (UK), FICA (in view)
Associate Editor, Risk and Financial Management Journal, USA
Editor, Journal of Economics and Management Sciences, USA
Former Associate Editor, Journal of Risk Model Validation, UK
PEER-REVIEWER, RISK.NET Journals, London
Former, Steering Committee Member, PRMIA Nigeria Chapter
Correspondence: Mathematics Programme Building, 118 National Mathematical Centre, Small Sheda, Kwali, 904105, FCT-Abuja, Nigeria. Tel: +2348038679094. E-mail: firstname.lastname@example.org
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Backward propagation and Activation Functions: