def f(x): return x**2 # Objective function def gradf(x): return 2 * x # Its derivative def gd(eta): x = 10 results = [x] for i in range(20): # eta 学习率 x -= eta * gradf(x) results.append(x) print('epoch 20, x:', x) return results res = gd(0.2) 梯度下降轨迹...
MHBF convex iff Hessian matrix positive semidefinite Hey! A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex if for all $x,y\in \mathbb{R}^n$ the inequality $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$ holds for all $t\in [0,1]$. Show that a twice continuously...
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However, the objective function and the constraint functions are convex in those papers. In this paper, we consider the constrained convex program (1.1). We convert it into an unconstrained optimization problem by using the exact penalty function. Then we show the objective function of this ...
and thus converge almost surely to a minimizer of the objective function. Moreover, we prove a general sublinear rate of convergence in expectation for the function values and a linear rate of convergence in expectation under an error bound condition of Tseng type. Under the same condition ...
In the second part of the paper, it is shown how the quadratic program may be solved as a coupled pair of primal and dual quadratic programs created from the original by simultaneously shifting the simple-bound constraints and adding a penalty term to the objective function. Any conventional ...
whereR(u) is regularization term that embodies the priors,F(f−u)is the fidelity term that forces the closeness of the restorationuto the observationf,Uis a function space modeling the restorationu, andλ>0is a tuning parameter that controls the tradeoff between the two terms. ...
The main idea is to verify whether a quadratic function constructed from the Hessian matrix of U(⋅) (the matrix of 2nd partial derivatives) is a Sum-of-Squares (see Supplement). We can then formulate the problem of finding a convex polynomial underestimator of the sample points (ϕ(i...
First we assume that we know the set of directions conjugate with respect to the hessian of the goal function. We applynsimultaneous directional minimizations along these conjugate directions starting from the same point followed by the addition of the directional corrections. Theorem justifying that ...
Three Operator Splitting with a Nonconvex Loss Function (2021) On the Oracle Complexity of Higher-Order Smooth Non-Convex Finite-Sum Optimization(2021) On the Theory of Implicit Deep Learning: Global Convergence with Implicit Layers (2021) Don’t Fix What ain’t Broke: Near-optimal Local Conver...