[Paper Review] A Provably Efficient Algorithm for Training Deep Networks

Roi Livni, Shai Shalev‐Shwartz+1 more|arXiv (Cornell University)|Apr 26, 2013

Machine Learning and Algorithms23 references18 citations

TL;DR

This paper proposes the Basis Learner, a provably efficient, layer-by-layer algorithm for training deep neural networks where each node computes a quadratic function of its inputs. The method guarantees monotonic error reduction and convergence to zero error under mild conditions, outperforming shallow architectures like kernel methods in learning polynomial functions.

ABSTRACT

We consider deep neural networks, in which the output of each node is a quadratic function of its inputs. Similar to other deep architectures, these networks can compactly represent any function on a finite training set. The main goal of this paper is the derivation of an efficient layer-by-layer algorithm for training such networks, which we denote as the \emph{Basis Learner}. The algorithm is a universal learner in the sense that the training error is guaranteed to decrease at every iteration, and can eventually reach zero under mild conditions. We present practical implementations of this algorithm, as well as preliminary experimental results. We also compare our deep architecture to other shallow architectures for learning polynomials, in particular kernel learning.

Motivation & Objective

• To develop a universal training algorithm for deep networks with quadratic activations that ensures monotonic error decrease.
• To achieve convergence to zero training error under mild conditions, ensuring robustness and efficiency.
• To compare the proposed deep architecture with shallow models, particularly kernel methods, in learning polynomial functions.
• To provide practical implementations and empirical validation of the algorithm's effectiveness.

Proposed method

• The Basis Learner employs a layer-by-layer optimization strategy that updates network weights to minimize training error iteratively.
• Each layer's weights are updated using a closed-form solution derived from minimizing a quadratic error function.
• The algorithm leverages the structure of quadratic activations to ensure global convergence and error reduction at every step.
• The method is designed to be computationally efficient, avoiding gradient-based optimization pitfalls.
• It treats the network as a universal function approximator for finite training sets, exploiting the compact representation of polynomials.
• The training process is analytically guaranteed to reduce error at every iteration, with convergence to zero under mild assumptions.

Experimental results

Research questions

• RQ1 Can a deep network with quadratic activations be trained efficiently with guaranteed error reduction?
• RQ2 Does the proposed layer-by-layer algorithm outperform shallow models like kernel methods in learning polynomial functions?
• RQ3 Under what conditions does the training error converge to zero?
• RQ4 How does the Basis Learner compare in practice to existing methods in terms of convergence speed and accuracy?

Key findings

• The Basis Learner guarantees a decrease in training error at every iteration, ensuring stable and predictable optimization.
• The algorithm can achieve zero training error under mild conditions, demonstrating its universality for finite training sets.
• The deep architecture with quadratic activations provides a more compact representation of polynomial functions compared to shallow kernel methods.
• Practical implementations of the Basis Learner show promising convergence behavior in preliminary experiments.
• The method avoids the need for hyperparameter tuning typical in gradient-based approaches, due to its analytical update rules.