Fundamentals Of Convex Analysis

By Leon Hirsch. 2016

774 citations recorded by the source corpus.

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In Mathematical Optimization

Citations in the corpus

f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization (2016, background): “Every convex, lower-semicontinuous function f has a convex conjugate function f∗, also known as Fenchel conjugate [15].”
CoCoA: A General Framework for Communication-Efficient Distributed Optimization (2016, background): “These cases follow since g ∗ i is µ -strongly convex if and only if g i is (1 /µ ) -smooth (Hiriart-Urruty and Lemaréchal, 2001, Theorem 4.2.2).”
On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning (2017, methodology): “Interested readers may refer to [38], [52] or any standard textbooks on convex analysis, for example, [7], [24], [28].”
The implicit bias of gradient descent on nonseparable data (2019, background): “It follows that the risk itself is 0-coercive (Hiriart-Urruty and Lemar´echal, 2001), and moreover strongly convex over bounded subsets, with a unique optimum ¯ v .”
On Fenchel Mini-Max Learning (2019, background): “Algorithm 1 Fenchel Mini-Max Learning Empirical data distribution p̂d = {xi}i=1 Proposal q(x), learning rate schedule {ηt} Initialize parameters θ, b for t = 1, 2, · · · do Sample {xt,j}j=1 ∼ p̂d(x), {xt,j}j=1 ∼ q(x) ut,j = ψ(xt,j) + b, It,j = exp(ψθ(xt,j)− ψθ(xt,j)− log q(xt,j)) Jt = ∑ j{ut,j + exp”
Stochastic Mirror Descent: Convergence Analysis and Adaptive Variants via the Mirror Stochastic Polyak Stepsize (2021, background): “If f is strictly convex and 1-coercive 6 then its conjugate function f ∗ is also strictly convex (Hiriart-Urruty & Lemaréchal, 2004)[Corollary 4.1.3] and we have et al., 1997)[Theorem 3.7], implying that the average gradients 1 / t t s =1 ∇ f ( x s ) or at least one of ∇ f ( x s ) converges to a…”
Stochastic Primal-Dual Coordinate Method for Regularized Empirical Risk Minimization (2014, background): “This leads to a convex-concave saddle point problem min Under Assumption A, each φ ∗ i is γ -strongly convex (since φ i is (1 /γ )-smooth; see, e.g., [14, Theorem 4.2.2]) and g is λ -strongly convex.”
Dual Discriminator Generative Adversarial Nets (2017, background): “This function has a convex conjugate function f∗, also known as Fenchel conjugate [13] : f∗ (t) = supu∈domf {ut− f (u)}.”
Learning from Conditional Distributions via Dual Embeddings (2016, background): “Furthermore, u∗(x, y) is related to the conditional distribution via u∗(x, y) ∈ ∂`y(Ez|x[f(z, x)]) [6].”
Discriminator Contrastive Divergence: Semi-Amortized Generative Modeling by Exploring Energy of the Discriminator (2020, methodology): “With r ( x ) = x and m ( x ) as the Fenchel conjugate[16] of a convex and lower-semicontinuous function, the objective in Eq.”
Risk and parameter convergence of logistic regression (2018, background): “Consequently, L◦A has compact sublevel sets over S (Hiriart-Urruty and Lemaréchal, 2001, Proposition B.3.2.4).”
Convex Optimization in Normed Spaces: Theory, Methods and Examples (2015, background): “The exposition in [64] is deep, well-written and rich in geometric insight (see also [26]).”
Kernel Exponential Family Estimation via Doubly Dual Embedding (2018, background): “Such a relationship is known as Fenchel duality (Rockafellar, 1970; Hiriart-Urruty and Lemar´echal, 2012).”
Curvature of Feasible Sets in Offline and Online Optimization (2020, background): “We need some basic notions from convex analysis, for which we refer to [24].”