  No-regret algorithms are popular for learning Nash equilibrium (NE) in
two-player zero-sum normal-form games (NFGs) and extensive-form games (EFGs).
Many recent works consider the last-iterate convergence no-regret algorithms.
Among them, the two most famous algorithms are Optimistic Gradient Descent
Ascent (OGDA) and Optimistic Multiplicative Weight Update (OMWU). However, OGDA
has high per-iteration complexity. OMWU exhibits a lower per-iteration
complexity but poorer empirical performance, and its convergence holds only
when NE is unique. Recent works propose a Reward Transformation (RT) framework
for MWU, which removes the uniqueness condition and achieves competitive
performance with OMWU. Unfortunately, RT-based algorithms perform worse than
OGDA under the same number of iterations, and their convergence guarantee is
based on the continuous-time feedback assumption, which does not hold in most
scenarios. To address these issues, we provide a closer analysis of the RT
framework, which holds for both continuous and discrete-time feedback. We
demonstrate that the essence of the RT framework is to transform the problem of
learning NE in the original game into a series of strongly convex-concave
optimization problems (SCCPs). We show that the bottleneck of RT-based
algorithms is the speed of solving SCCPs. To improve the their empirical
performance, we design a novel transformation method to enable the SCCPs can be
solved by Regret Matching+ (RM+), a no-regret algorithm with better empirical
performance, resulting in Reward Transformation RM+ (RTRM+). RTRM+ enjoys
last-iterate convergence under the discrete-time feedback setting. Using the
counterfactual regret decomposition framework, we propose Reward Transformation
CFR+ (RTCFR+) to extend RTRM+ to EFGs. Experimental results show that our
algorithms significantly outperform existing last-iterate convergence
algorithms and RM+ (CFR+).
