What is Perceptron ?
Perceptron is a probabilistic model for brain information storage and organizing. It takes an empiricist, or “connectionist,” view of information retention and action. This contradicts notions that information is kept as coded representations or images, which would require a one-to-one mapping between sensory stimuli and recorded pattern. Connectionists believe retention occurs through new neural system pathways or functionally comparable situations. These new pathways instantly trigger the right reaction to new stimuli without requiring recognition or identification.
The Perceptron model was created to demonstrate some basic aspects of intelligent systems without getting into the unique, often unknown, conditions of specific biological creatures. The Perceptron uses probability theory to analyze systems with an unknown structure, unlike symbolic logic and idealized wiring diagram models, which focus on how a deterministic system might perform perception and recall. The hypothesis is based on statistical separability.
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Perceptron Algorithm
Inputs:
- Training data: {(x1,y1),(x2,y2),…,(xm,ym)}\{(x_1, y_1), (x_2, y_2), \dots, (x_m, y_m)\}{(x1,y1),(x2,y2),…,(xm,ym)}, where:
- xi∈Rnx_i \in \mathbb{R}^nxi∈Rn: feature vector
- yi∈{−1,+1}y_i \in \{-1, +1\}yi∈{−1,+1}: label
- Learning rate η>0\eta > 0η>0
- Number of epochs TTT
Initialize:
- Weights: w=0∈Rnw = 0 \in \mathbb{R}^nw=0∈Rn
- Bias: b=0b = 0b=0
Algorithm:
For t = 1 to T:
For each training example (x_i, y_i):
Compute activation: a = w · x_i + b
Predict: y_pred = sign(a)
If y_pred ≠ y_i:
Update weights: w = w + η * y_i * x_i
Update bias: b = b + η * y_i
Output:
- Final weight vector www
- Bias bbb
Key Perceptron assumptions from Hebb, Hayek, Uttley, and Ashby
- Physical connections involved in learning and recognition differ across creatures and are random at birth with little genetic limitations.
- The system is malleable because brain activity can permanently remodel neurons, changing the likelihood that a stimulus in one set of cells will induce a response in another.
- After several stimuli, “similar” cells develop connections to the same sets of responsive cells, whereas “dissimilar” cells connect to distinct sets.
- Positive and negative reinforcement can help or impede connection-making.
- Similar stimuli activate similar nervous system cells. It depends on the physical organization of the perceiving system, which grows through environmental interaction.
Perceptron organization:
Photo-perceptrons that respond to optical patterns are layered. In certain models, retinal S-points answer all-or-nothing.
S-points (origin points) send excitatory or inhibitory impulses to association cells (A-units) in a “projection area” (Ai). When the sum of impulses reaches its threshold (θ), an A-unit fires. The projection area has clustered or focalized A-unit origin points. In some versions, the retina connects directly to the association area without this projection area.
Assume random connections between the projection region (Ai) and the primary association area (An). An’s A-units acquire fibers from random Ai origins and function similarly.
Response units (Ri, R2,…, Rn) are A-unit-like cells. In the source-set for each response, a significant number of origin points are randomly located in An.
The An layer receives forward connections. The An and R-units have two-way feedback connections. Feedback rules can be:
Each response has excitatory feedback from its source-set cells.
Each response inhibits untransmitting association cells by feeding back to its source-set.
A simplified 3-stage perceptron (S -> An -> R) analyzes coincident stimulus regions rather than contours to derive similarity notions. Since one reaction inhibits the other and its source-set, they are mutually incompatible in this reduced system. A-units or their connections must be modified so that stimuli of one class trigger a larger impulse in one source-set than another for the system to learn.
Learning revolves around “value”. Value is added to active A-units. This improves the likelihood of the same reaction when the stimulus is offered again, suggesting learning. This system’s single hypothetical variable (construct) is value, which must meet functional requirements and have a potentially measurable physical correlation.
Phases of stimuli response
A portion of A-units are triggered in the main phase. The postdominant phase suppresses others by limiting activity to a single source-set. Initial dominance is random, but reinforced A-units gain value and become dominant upon stimulus re-presentation.
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The dominant phase analysis comprises two variables:
Pa: Expected A-unit activation per stimulus size. Consider the proportion of lit S-points (R), number of excitatory and inhibitory connections (x, y), and A-unit threshold (θ). Increase the threshold or inhibitory connections to lower Pa.
Pc: The conditional chance that an A-unit reacting to one stimulus will respond to another. As threshold and inhibitory connections rise, it falls. Even with disjunct stimuli, Pc is greater than zero. Pc unites as stimuli match28. Lower Pc for smaller stimuli and higher thresholds.
Mathematical study of learning evaluates the system’s performance in terms of Pr (probability of accurate response between two options after a learning period) and Pg. Two systems are analyzed to determine the prevailing response: the mean-discriminating (μ-system) and the sum-discriminating (Σ-system).
A general equation approximates Pr and Pg. This equation incorporates elements such as P(Nar > 0) and constants (c1, c2, c3, c4) based on perceptron physical properties and stimulus environment organization.
Performance of the perceptron in several environments
Ideal Environment: Random lighting points without intrinsic similarity categorization. In this scenario, c1 is zero, hence Pg cannot exceed the random expectation. Pr analysis shows that alpha (α), beta (β), and gamma (γ) systems have distinct value-gain mechanisms. The μ-system typically outperforms the Σ-system.
Differentiated Environment: Multiple stimulus classes exist. When each response’s stimuli are from a different class, c1 becomes non-zero. This makes the performance asymptote non-random. Interestingly, Pg approaches Pr’s asymptote in the differentiated environment. In the limit, performance is good whether the test stimulus has been observed before or not. Better-than-chance performance occurs when the predicted value of Pc between members of the same class (Pc11) is larger than Pa, which is greater than Pc between stimuli from different classes (Pc12).
Bivalent systems offer positive and negative reinforcement. These systems allow active A-units to gain or lose value based on reinforcement signals. Bivalent systems reduce bias better. They facilitate trial-and-error learning. Computer simulations confirmed the theory’s predictions for bivalent systems.
Spontaneous Organization and Better Perceptrons
- If stimuli leave a record, statistical separability can separate temporal patterns like velocities or sound sequences.
- In the projection area, constraining origin point spatial distribution can make A-units contour-sensitive and increase performance.
- A-unit decay allows “spontaneous” idea creation, when the perceptron distinguishes stimulus classes without reinforcement.