Rate-Preserving Reductions for Blackwell Approachability

TL;DR

This work analyzes the rate-preserving relationship between Blackwell approachability and no-regret learning, showing that Abernethy et al.'s classical reduction can fail to preserve optimal rates. It introduces improper $\phi$-regret minimization as a tight, rate-preserving target for reducing any approachability instance, and provides a precise linear-equivalence framework to compare regret minimization classes. The authors characterize when improper $\phi$-regret reduces to standard classes (external or proper $\phi$-regret) and prove that some improper instances cannot be so reduced, implying approachability captures problems beyond standard online learning. They also offer an algorithmic test to decide reducibility in the polytopal setting and discuss weighted regret as a motivating warm-up. Overall, the paper deepens the understanding of when rate information is preserved under reductions and highlights the nuanced landscape linking approachability with regret minimization.

Abstract

Abernethy et al. (2011) showed that Blackwell approachability and no-regret learning are equivalent, in the sense that any algorithm that solves a specific Blackwell approachability instance can be converted to a sublinear regret algorithm for a specific no-regret learning instance, and vice versa. In this paper, we study a more fine-grained form of such reductions, and ask when this translation between problems preserves not only a sublinear rate of convergence, but also preserves the optimal rate of convergence. That is, in which cases does it suffice to find the optimal regret bound for a no-regret learning instance in order to find the optimal rate of convergence for a corresponding approachability instance? We show that the reduction of Abernethy et al. (2011) does not preserve rates: their reduction may reduce a $d$-dimensional approachability instance $I_1$ with optimal convergence rate $R_1$ to a no-regret learning instance $I_2$ with optimal regret-per-round of $R_2$, with $R_{2}/R_{1}$ arbitrarily large (in particular, it is possible that $R_1 = 0$ and $R_{2} > 0$). On the other hand, we show that it is possible to tightly reduce any approachability instance to an instance of a generalized form of regret minimization we call improper $φ$-regret minimization (a variant of the $φ$-regret minimization of Gordon et al. (2008) where the transformation functions may map actions outside of the action set). Finally, we characterize when linear transformations suffice to reduce improper $φ$-regret minimization problems to standard classes of regret minimization problems in a rate preserving manner. We prove that some improper $φ$-regret minimization instances cannot be reduced to either subclass of instance in this way, suggesting that approachability can capture some problems that cannot be phrased in the language of online learning.

Figures (4)

• Figure 1: Overview of problem classes and reductions. AbernethyBartlettHazan2011 give a non-tight reduction between approachability and external regret minimization. We give in Theorem \ref{['thm:app-to-improper']} a tight (rate-preserving) reduction between approachability and the class of improper $\phi$-regret minimization problems. Further, we characterize when an improper $\phi$-regret instance is tightly reducible, under linear equivalence, to more well-studied classes of regret minimization problems, like proper $\phi$-regret minimization (Theorem \ref{['thm:wreg-reduction']}) and external regret minimization (Theorem \ref{['thm:external-char']}).
• Figure 2: Examples of improper $\phi$-regret minimization
• Figure 3: Illustration of the definition of $S$ in the proof of Lemma \ref{['lemma:one-phi-external']}. $v_1, v_2, v_3$ represent the vertices of the simplex in $\mathbb{R}^3$ and $p$ is the fixed point of $\phi$. The vector $\phi(v_k)$, $k \in [3]$, may not be in the simplex. The linear function $S$ maps vector $(\phi - I)(v_k)$ into $p - v_k$.
• Figure 4: For any $t$, the matrix $M(t) = A + tB$ has a right kernel element of the form $(x(t), y(t), 1)$, where $x(t) = \frac{72t^2-46t + 81}{2(21t^2 + 8t + 5)}$ and $y(t) = \frac{72t^2 - 41t + 36}{21t^2 + 8t + 5}$. The above diagram plots $(x(t), y(t))$ for all $t \in \mathbb{R}$, and shows that this right kernel element always has non-negative entries (and hence has a multiple that lies in $\Delta_3$).

Theorems & Definitions (28)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof