85 research outputs found
From Proximal Point Method to Nesterov's Acceleration
The proximal point method (PPM) is a fundamental method in optimization that
is often used as a building block for fast optimization algorithms. In this
work, building on a recent work by Defazio (2019), we provide a complete
understanding of Nesterov's accelerated gradient method (AGM) by establishing
quantitative and analytical connections between PPM and AGM. The main
observation in this paper is that AGM is in fact equal to a simple
approximation of PPM, which results in an elementary derivation of the
mysterious updates of AGM as well as its step sizes. This connection also leads
to a conceptually simple analysis of AGM based on the standard analysis of PPM.
This view naturally extends to the strongly convex case and also motivates
other accelerated methods for practically relevant settings.Comment: 14 pages; Section 4 updated; Remark 5 added; comments would be
appreciated
Connections Between Adaptive Control and Optimization in Machine Learning
This paper demonstrates many immediate connections between adaptive control
and optimization methods commonly employed in machine learning. Starting from
common output error formulations, similarities in update law modifications are
examined. Concepts in stability, performance, and learning, common to both
fields are then discussed. Building on the similarities in update laws and
common concepts, new intersections and opportunities for improved algorithm
analysis are provided. In particular, a specific problem related to higher
order learning is solved through insights obtained from these intersections.Comment: 18 page
On the Stability and Convergence of Stochastic Gradient Descent with Momentum
While momentum-based methods, in conjunction with the stochastic gradient
descent, are widely used when training machine learning models, there is little
theoretical understanding on the generalization error of such methods. In
practice, the momentum parameter is often chosen in a heuristic fashion with
little theoretical guidance. In the first part of this paper, for the case of
general loss functions, we analyze a modified momentum-based update rule, i.e.,
the method of early momentum, and develop an upper-bound on the generalization
error using the framework of algorithmic stability. Our results show that
machine learning models can be trained for multiple epochs of this method while
their generalization errors are bounded. We also study the convergence of the
method of early momentum by establishing an upper-bound on the expected norm of
the gradient. In the second part of the paper, we focus on the case of strongly
convex loss functions and the classical heavy-ball momentum update rule. We use
the framework of algorithmic stability to provide an upper-bound on the
generalization error of the stochastic gradient method with momentum. We also
develop an upper-bound on the expected true risk, in terms of the number of
training steps, the size of the training set, and the momentum parameter.
Experimental evaluations verify the consistency between the numerical results
and our theoretical bounds and the effectiveness of the method of early
momentum for the case of non-convex loss functions
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