The sample complexity of pattern classification with neural networks: the size of the weights is more important than the size of the network

Sample complexity results from computational learning theory, when applied to neural network learning for pattern classification problems, suggest that for good generalization performance the number of training examples should grow at least linearly with the number of adjustable parameters in the network. Results in this paper show that if a large neural network is used for a pattern classification problem and the learning algorithm finds a network with small weights that has small squared error on the training patterns, then the generalization performance depends on the size of the weights rather than the number of weights. For example, consider a two-layer feedforward network of sigmoid units, in which the sum of the magnitudes of the weights associated with each unit is bounded by A and the input dimension is n. We show that the misclassification probability is no more than a certain error estimate (that is related to squared error on the training set) plus A3 √((log n)/m) (ignoring log A and log m factors), where m is the number of training patterns. This may explain the generalization performance of neural networks, particularly when the number of training examples is considerably smaller than the number of weights. It also supports heuristics (such as weight decay and early stopping) that attempt to keep the weights small during training. The proof techniques appear to be useful for the analysis of other pattern classifiers: when the input domain is a totally bounded metric space, we use the same approach to give upper bounds on misclassification probability for classifiers with decision boundaries that are far from the training examples.

Primary and secondary neural networks of auditory prepulse inhibition: a functional magnetic resonance imaging study of sensorimotor gating of the human acoustic startle response

Feedforward inhibition deficits have been consistently demonstrated in a range of neuropsychiatric conditions using prepulse inhibition (PPI) of the acoustic startle eye-blink reflex when assessing sensorimotor gating. While PPI can be recorded in acutely decerebrated rats, behavioural, pharmacological and psychophysiological studies suggest the involvement of a complex neural network extending from brainstem nuclei to higher order cortical areas. The current functional magnetic resonance imaging study investigated the neural network underlying PPI and its association with electromyographically (EMG) recorded PPI of the acoustic startle eye-blink reflex in 16 healthy volunteers. A sparse imaging design was employed to model signal changes in blood oxygenation level-dependent (BOLD) responses to acoustic startle probes that were preceded by a prepulse at 120 ms or 480 ms stimulus onset asynchrony or without prepulse. Sensorimotor gating was EMG confirmed for the 120-ms prepulse condition, while startle responses in the 480-ms prepulse condition did not differ from startle alone. Multiple regression analysis of BOLD contrasts identified activation in pons, thalamus, caudate nuclei, left angular gyrus and bilaterally in anterior cingulate, associated with EMGrecorded sensorimotor gating. Planned contrasts confirmed increased pons activation for startle alone vs 120-ms prepulse condition, while increased anterior superior frontal gyrus activation was confirmed for the reverse contrast. Our findings are consistent with a primary pontine circuitry of sensorimotor gating that interconnects with inferior parietal, superior temporal, frontal and prefrontal cortices via thalamus and striatum. PPI processes in the prefrontal, frontal and superior temporal cortex were functionally distinct from sensorimotor gating.

Turbocharging Networks in the Public Sector

Networks have come to occupy a key position in the strategic armoury of the government, business and community sectors and now have impact on a broad array of policy and management arenas. An emphasis on relationships, trust and mutuality mean that networks function on a different operating logic to the conventional processes of government and business. It is therefore important that organizational members of networks are able to adopt the skills and culture necessary to operate successfully under these distinctive kinds of arrangements. Because networks function from a different operational logic to traditional bureaucracies, public sector organizations may experience difficulties in adapting to networked arrangements. Networks are formed to address a variety of social problems or meet capability gaps within organizations. As such they are often under pressure to quickly produce measurable outcomes and need to form rapidly and come to full operation quickly. This paper presents a theoretical exploration of how diverse types of networks are required for different management and policy situations and draws on a set of public sector case studies to understand/demonstrate how these various types of networked arrangements may be ‘turbo-charged’ so that they more quickly adopt the characteristics necessary to deliver required outcomes.