13 research outputs found
Strict Monotonicity and Convergence Rate of Titterington's Algorithm for Computing D-optimal Designs
We study a class of multiplicative algorithms introduced by Silvey et al.
(1978) for computing D-optimal designs. Strict monotonicity is established for
a variant considered by Titterington (1978). A formula for the rate of
convergence is also derived. This is used to explain why modifications
considered by Titterington (1978) and Dette et al. (2008) usually converge
faster
Critical Slowing Down Near Topological Transitions in Rate-Distortion Problems
In Rate Distortion (RD) problems one seeks reduced representations of a
source that meet a target distortion constraint. Such optimal representations
undergo topological transitions at some critical rate values, when their
cardinality or dimensionality change. We study the convergence time of the
Arimoto-Blahut alternating projection algorithms, used to solve such problems,
near those critical points, both for the Rate Distortion and Information
Bottleneck settings. We argue that they suffer from Critical Slowing Down -- a
diverging number of iterations for convergence -- near the critical points.
This phenomenon can have theoretical and practical implications for both
Machine Learning and Data Compression problems.Comment: 9 pages, 2 figures, ISIT 2021 submissio
Analytical calculation formulas for capacities of classical and classical-quantum channels
We derive an analytical calculation formula for the channel capacity of a
classical channel without any iteration while its existing algorithms require
iterations and the number of iteration depends on the required precision level.
Hence, our formula is its first analytical formula without any iteration. We
apply the obtained formula to examples and see how the obtained formula works
in these examples. Then, we extend it to the channel capacity of a
classical-quantum (cq-) channel. Many existing studies proposed algorithms for
a cq-channel and all of them require iterations. Our extended analytical
algorithm have also no iteration and output the exactly optimum values
A Constrained BA Algorithm for Rate-Distortion and Distortion-Rate Functions
The Blahut-Arimoto (BA) algorithm has played a fundamental role in the
numerical computation of rate-distortion (RD) functions. This algorithm
possesses a desirable monotonic convergence property by alternatively
minimizing its Lagrangian with a fixed multiplier. In this paper, we propose a
novel modification of the BA algorithm, wherein the multiplier is updated
through a one-dimensional root-finding step using a monotonic univariate
function, efficiently implemented by Newton's method in each iteration.
Consequently, the modified algorithm directly computes the RD function for a
given target distortion, without exploring the entire RD curve as in the
original BA algorithm. Moreover, this modification presents a versatile
framework, applicable to a wide range of problems, including the computation of
distortion-rate (DR) functions. Theoretical analysis shows that the outputs of
the modified algorithms still converge to the solutions of the RD and DR
functions with rate , where is the number of iterations.
Additionally, these algorithms provide -approximation solutions
with
arithmetic operations, where are the sizes of source and reproduced
alphabets respectively. Numerical experiments demonstrate that the modified
algorithms exhibit significant acceleration compared with the original BA
algorithms and showcase commendable performance across classical source
distributions such as discretized Gaussian, Laplacian and uniform sources.Comment: Version_
Deciding What to Model: Value-Equivalent Sampling for Reinforcement Learning
The quintessential model-based reinforcement-learning agent iteratively
refines its estimates or prior beliefs about the true underlying model of the
environment. Recent empirical successes in model-based reinforcement learning
with function approximation, however, eschew the true model in favor of a
surrogate that, while ignoring various facets of the environment, still
facilitates effective planning over behaviors. Recently formalized as the value
equivalence principle, this algorithmic technique is perhaps unavoidable as
real-world reinforcement learning demands consideration of a simple,
computationally-bounded agent interacting with an overwhelmingly complex
environment, whose underlying dynamics likely exceed the agent's capacity for
representation. In this work, we consider the scenario where agent limitations
may entirely preclude identifying an exactly value-equivalent model,
immediately giving rise to a trade-off between identifying a model that is
simple enough to learn while only incurring bounded sub-optimality. To address
this problem, we introduce an algorithm that, using rate-distortion theory,
iteratively computes an approximately-value-equivalent, lossy compression of
the environment which an agent may feasibly target in lieu of the true model.
We prove an information-theoretic, Bayesian regret bound for our algorithm that
holds for any finite-horizon, episodic sequential decision-making problem.
Crucially, our regret bound can be expressed in one of two possible forms,
providing a performance guarantee for finding either the simplest model that
achieves a desired sub-optimality gap or, alternatively, the best model given a
limit on agent capacity.Comment: Accepted to Neural Information Processing Systems (NeurIPS) 202