5,559 research outputs found
The Computational Power of Optimization in Online Learning
We consider the fundamental problem of prediction with expert advice where
the experts are "optimizable": there is a black-box optimization oracle that
can be used to compute, in constant time, the leading expert in retrospect at
any point in time. In this setting, we give a novel online algorithm that
attains vanishing regret with respect to experts in total
computation time. We also give a lower bound showing
that this running time cannot be improved (up to log factors) in the oracle
model, thereby exhibiting a quadratic speedup as compared to the standard,
oracle-free setting where the required time for vanishing regret is
. These results demonstrate an exponential gap between
the power of optimization in online learning and its power in statistical
learning: in the latter, an optimization oracle---i.e., an efficient empirical
risk minimizer---allows to learn a finite hypothesis class of size in time
. We also study the implications of our results to learning in
repeated zero-sum games, in a setting where the players have access to oracles
that compute, in constant time, their best-response to any mixed strategy of
their opponent. We show that the runtime required for approximating the minimax
value of the game in this setting is , yielding
again a quadratic improvement upon the oracle-free setting, where
is known to be tight
On-line PCA with Optimal Regrets
We carefully investigate the on-line version of PCA, where in each trial a
learning algorithm plays a k-dimensional subspace, and suffers the compression
loss on the next instance when projected into the chosen subspace. In this
setting, we analyze two popular on-line algorithms, Gradient Descent (GD) and
Exponentiated Gradient (EG). We show that both algorithms are essentially
optimal in the worst-case. This comes as a surprise, since EG is known to
perform sub-optimally when the instances are sparse. This different behavior of
EG for PCA is mainly related to the non-negativity of the loss in this case,
which makes the PCA setting qualitatively different from other settings studied
in the literature. Furthermore, we show that when considering regret bounds as
function of a loss budget, EG remains optimal and strictly outperforms GD.
Next, we study the extension of the PCA setting, in which the Nature is allowed
to play with dense instances, which are positive matrices with bounded largest
eigenvalue. Again we can show that EG is optimal and strictly better than GD in
this setting
Adaptive Momentum for Neural Network Optimization
In this thesis, we develop a novel and efficient algorithm for optimizing neural networks inspired by a recently proposed geodesic optimization algorithm. Our algorithm, which we call Stochastic Geodesic Optimization (SGeO), utilizes an adaptive coefficient on top of Polyaks Heavy Ball method effectively controlling the amount of weight put on the previous update to the parameters based on the change of direction in the optimization path. Experimental results on strongly convex functions with Lipschitz gradients and deep Autoencoder benchmarks show that SGeO reaches lower errors than established first-order methods and competes well with lower or similar errors to a recent second-order method called K-FAC (Kronecker-Factored Approximate Curvature). We also incorporate Nesterov style lookahead gradient into our algorithm (SGeO-N) and observe notable improvements. We believe that our research will open up new directions for high-dimensional neural network optimization where combining the efficiency of first-order methods and the effectiveness of second-order methods proves a promising avenue to explore
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