ora-0163

14.5 Tangent repair, stability, and convergence

A tangent category axiomatizes coherent infinitesimal transport; it does not, by itself, specify when an infinite trajectory converges. Convergence therefore has three distinct layers:

base convergence, such as \(x_t\to x_\star \);
tangent stability, meaning that the variational trajectory \(\dot x_t\) is bounded or convergent; and
limit compatibility, meaning that differentiating the parameterized limit agrees with the limit of the tangents.

None follows formally from the other two. In an abstract tangent category, the third layer requires an additional convergence doctrine—for example, a topology or metric enrichment with specified sequential limits—and a requirement that the tangent functor preserve the relevant limits. In the finite-dimensional smooth realization used here, these questions reduce to ordinary continuity and stability estimates.

Theorem 14.6 Normalized quadratic FTRL convergence lift

Let \(a_t{\gt}0\), and write

\[ \bar H_t=H_t/a_t, \qquad \bar q_t=q_t/a_t. \]

Suppose a \(C^1\) parameterized family satisfies

\[ \bar H_t\to H_\star \succ 0, \qquad \bar q_t\to q_\star , \]

and, in a parameter direction \(v\),

\[ D\bar H_t[v]\to \dot H_\star , \qquad D\bar q_t[v]\to \dot q_\star . \]

Assume also that \(\lambda _{\min }(\bar H_t)\geq \mu {\gt}0\) uniformly. Then the FTRL decisions and their tangent decisions converge:

\[ x_{t+1}=-\bar H_t^{-1}\bar q_t \longrightarrow x_\star =-H_\star ^{-1}q_\star , \]

\[ Dx_{t+1}[v] \longrightarrow -H_\star ^{-1}(\dot q_\star +\dot H_\star x_\star ). \]

Consequently, whenever the parameterized limits above hold locally with enough uniformity to differentiate the limit, the tangent comparison with the asymptotic decision is exact:

\[ D\! \left(\lim _{t\to \infty }x_{t+1}\right)[v] = \lim _{t\to \infty }Dx_{t+1}[v]. \]

Proof ▼

Uniform coercivity bounds the inverse operators: \(\lVert \bar H_t^{-1}\rVert \leq \mu ^{-1}\). Continuity of inversion on the positive-definite cone gives \(\bar H_t^{-1}\to H_\star ^{-1}\), and hence the base decisions converge. Differentiating \(\bar H_tx_{t+1}+\bar q_t=0\) gives

\[ Dx_{t+1}[v] =-\bar H_t^{-1} \bigl(D\bar q_t[v]+D\bar H_t[v]x_{t+1}\bigr). \]

Every factor on the right converges, yielding the displayed tangent limit. The final equality is precisely the additional local uniformity assumption, not a consequence of the tangent-category axioms alone.

The normalization is essential: cumulative sufficient statistics commonly grow with \(t\), while their ratios can converge. The theorem is stronger than a regret statement in one respect and weaker in another: it controls the decision and its sensitivity, but it assumes convergence of normalized first-order data. A sublinear-regret bound alone need not imply convergence of either sequence.

The corresponding general mechanism is stability of the linearized update.

Theorem 14.7 Contractive tangent lift for smooth updates

Let \(z_{t+1}=U_t(z_t,\theta )\) be a finite-dimensional \(C^1\) update, and fix a parameter direction \(v\). Suppose \(z_t\to z_\star \),

\[ A_t:=D_zU_t(z_t,\theta )\to A_\star , \qquad b_t:=D_\theta U_t(z_t,\theta )[v]\to b_\star , \]

and \(\lVert A_t\rVert \leq q{\lt}1\) uniformly in one operator norm. Then the tangent recursion

\[ \dot z_{t+1}=A_t\dot z_t+b_t \]

converges to the unique fixed tangent

\[ \dot z_\star =(I-A_\star )^{-1}b_\star . \]

Proof ▼

The uniform contraction implies \(I-A_\star \) is invertible. With \(e_t=\dot z_t-\dot z_\star \),

\[ e_{t+1}=A_te_t+(A_t-A_\star )\dot z_\star +(b_t-b_\star ). \]

Thus \(\lVert e_{t+1}\rVert \leq q\lVert e_t\rVert +r_t\), where \(r_t\to 0\). Iterating this inequality and splitting the resulting geometric convolution into a finite prefix and a uniformly small tail gives \(e_t\to 0\).

Theorems 14.6 and 14.7 expose the algorithm-dependent boundary. Mirror descent can inherit a tangent convergence theorem when its update is contractive in the geometry induced by a uniformly strongly convex and smooth mirror potential. Equivalence with FTRL can transfer a proved result only when the equivalence itself holds for the parameterized family and its tangents. For Adam, the state must include the decision and both moment estimates; its tangent is the variational dynamics of that full state. Vanilla Adam has convex examples on which the base algorithm fails to converge [ Reddi et al. , 2019 ] , so no unconditional tangent convergence lift is possible. Convergent variants can be treated only after their full-state cocycle is shown stable; the usual \(\epsilon \) term also keeps the square-root readout smooth away from singular coordinate behavior.

Finally, one cannot obtain these results merely by differentiating a regret inequality. A pointwise statement \(R_T(\theta )\leq B_T(\theta )\) does not imply an inequality between their derivatives. Such a step requires a uniform parameterized bound, differentiability or directional regularity, and control of the parameter-dependent comparator.