Mirror descent and geometry

Choose a differentiable strictly convex generator ψ on the interior of its domain. The Bregman divergence from x to z is the gap between ψ(z) and the tangent plane to ψ at x. Convexity makes this nonnegative. It generally depends on argument order and need not obey a triangle inequality, so it is a divergence rather than a metric. Strong convexity relative to a chosen norm supplies the quantitative lower bound needed in many convergence analyses.

For $\psi(x)=\|x\|^2/2$, the divergence is half the squared Euclidean distance. Other choices reshape how movement is penalized. The choice affects both the gradient's relevant dual norm and the distance to the comparator in regret or convergence bounds. Geometry is therefore not just a decorative coordinate transformation: it enters the assumptions and constants of the method. One must also ensure that iterates remain in the generator's domain.

On the probability simplex, use negative entropy $\psi(p)=\sum_i p_i\log p_i$ and start with strictly positive coordinates. The resulting divergence is $\sum_i q_i\log(q_i/p_i)$ for probability vectors, with the usual continuous convention for zero terms. The normalization constraint couples coordinates. Solving the mirror subproblem gives weights proportional to the old weights multiplied by exponentials of negative scaled gradient components.

A coordinate with a larger cost gradient loses relative weight, while all weights are normalized to sum to one. In exact arithmetic positive initial weights remain positive under finite gradients. A coordinate initialized at zero cannot generally be revived by the multiplicative formula, which is one reason positive initialization matters. Numerically, exponentials can underflow; stable implementations use shifted log-weights and log-sum-exp normalization rather than assuming symbolic positivity prevents floating-point failure.

Euclidean projection and entropic mirror descent can both maintain simplex feasibility, yet their paths and guarantees use different movement costs. On some high-dimensional probability problems, entropy aligns well with bounded coordinatewise gradients and yields favorable dimension dependence. This is conditional on the norm bounds and generator properties; mirror descent is not uniformly faster for every objective. Choosing a geometry requires identifying the feasible set, gradient scale, and computational cost of its subproblem.

The lab is the Euclidean special case on the real line with no simplex constraint. Its fixed quadratic objective and step-size control let you inspect the baseline recurrence, not an entropy-generated update. Predict a Euclidean step there, then compare it with the manually normalized probability update below. No change of the lab's initial position turns it into a probability vector, and its Play control does not execute multiplicative weights or adapt the underlying geometry.