64 research outputs found
Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks
Stochastic gradient descent (SGD) is widely believed to perform implicit
regularization when used to train deep neural networks, but the precise manner
in which this occurs has thus far been elusive. We prove that SGD minimizes an
average potential over the posterior distribution of weights along with an
entropic regularization term. This potential is however not the original loss
function in general. So SGD does perform variational inference, but for a
different loss than the one used to compute the gradients. Even more
surprisingly, SGD does not even converge in the classical sense: we show that
the most likely trajectories of SGD for deep networks do not behave like
Brownian motion around critical points. Instead, they resemble closed loops
with deterministic components. We prove that such "out-of-equilibrium" behavior
is a consequence of highly non-isotropic gradient noise in SGD; the covariance
matrix of mini-batch gradients for deep networks has a rank as small as 1% of
its dimension. We provide extensive empirical validation of these claims,
proven in the appendix
Game theoretic controller synthesis for multi-robot motion planning Part I : Trajectory based algorithms
We consider a class of multi-robot motion planning problems where each robot
is associated with multiple objectives and decoupled task specifications. The
problems are formulated as an open-loop non-cooperative differential game. A
distributed anytime algorithm is proposed to compute a Nash equilibrium of the
game. The following properties are proven: (i) the algorithm asymptotically
converges to the set of Nash equilibrium; (ii) for scalar cost functionals, the
price of stability equals one; (iii) for the worst case, the computational
complexity and communication cost are linear in the robot number
Model Zoo: A Growing "Brain" That Learns Continually
This paper argues that continual learning methods can benefit by splitting
the capacity of the learner across multiple models. We use statistical learning
theory and experimental analysis to show how multiple tasks can interact with
each other in a non-trivial fashion when a single model is trained on them. The
generalization error on a particular task can improve when it is trained with
synergistic tasks, but can also deteriorate when trained with competing tasks.
This theory motivates our method named Model Zoo which, inspired from the
boosting literature, grows an ensemble of small models, each of which is
trained during one episode of continual learning. We demonstrate that Model Zoo
obtains large gains in accuracy on a variety of continual learning benchmark
problems
Incremental sampling based algorithms for state estimation
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2012.Cataloged from department-submitted PDF version of thesis. This electronic version was submitted and approved by the author's academic department as part of an electronic thesis pilot project. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 95-98).Perception is a crucial aspect of the operation of autonomous vehicles. With a multitude of different sources of sensor data, it becomes important to have algorithms which can process the available information quickly and provide a timely solution. Also, an inherently continuous world is sensed by robot sensors and converted into discrete packets of information. Algorithms that can take advantage of this setup, i.e., which have a sound founding in continuous time formulations but which can effectively discretize the available information in an incremental manner according to different requirements can potentially outperform conventional perception frameworks. Inspired from recent results in motion planning algorithms, this thesis aims to address these two aspects of the problem of robot perception, through novel incremental and anytime algorithms. The first part of the thesis deals with algorithms for different estimation problems, such as filtering, smoothing, and trajectory decoding. They share the basic idea that a general continuous-time system can be approximated by a sequence of discrete Markov chains that converge in a suitable sense to the original continuous time stochastic system. This discretization is obtained through intuitive rules motivated by physics and is very easy to implement in practice. Incremental algorithms for the above problems can then be formulated on these discrete systems whose solutions converge to the solution of the original problem. A similar construction is used to explore control of partially observable processes in the latter part of the thesis. A general continuous time control problem in this case is approximates by a sequence of discrete partially observable Markov decision processes (POMDPs), in such a way that the trajectories of the POMDPs -- i.e., the trajectories of beliefs -- converge to the trajectories of the original continuous problem. Modern point-based solvers are used to approximate control policies for each of these discrete problems and it is shown that these control policies converge to the optimal control policy of the original problem in an appropriate space. This approach is promising because instead of solving a large POMDP problem from scratch, which is PSPACE-hard, approximate solutions of smaller problems can be used to guide the search for the optimal control policy.by Pratik Chaudhari.S.M
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